http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Yjin&feedformat=atomSoft-Matter - User contributions [en]2020-08-05T23:18:52ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Yuhang_Jin&diff=23364Yuhang Jin2011-12-09T05:55:44Z<p>Yjin: </p>
<hr />
<div>Wiki entries:<br />
<br />
[[Hydrophilic PDMS microchannels for high-throughput formation of oil-in-water microdroplets and water-in-oil-in-water double emulsions.]]<br />
<br />
[[Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device.]]<br />
<br />
[[Paper on a disc: balancing the capillary-driven flow with a centrifugal force]]<br />
<br />
[[Two-dimensional nanometric confinement of entangled polymer melts]]<br />
<br />
[[Restructuring of Hydrophobic Surfaces Created by Surfactant Adsorption to Mica Surfaces]]<br />
<br />
[[The Phase Behavior of a Polymer-Fullerene Bulk Heterojunction System that Contains Bimolecular Crystals]]<br />
<br />
[[Concentration of Magnetic Beads Utilizing Light-Induced Electro-Osmosis Flow]]<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]<br />
<br />
[[Amphiphilic]]<br />
<br />
[[Centrifugal forces]]<br />
<br />
[[Coulomb interaction]]<br />
<br />
[[Laplace pressure]]<br />
<br />
[[Photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Yuhang_Jin&diff=23363Yuhang Jin2011-12-09T05:55:37Z<p>Yjin: </p>
<hr />
<div>Wiki entries:<br />
<br />
[[Hydrophilic PDMS microchannels for high-throughput formation of oil-in-water microdroplets and water-in-oil-in-water double emulsions.]]<br />
<br />
[[Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device.]]<br />
<br />
[[Paper on a disc: balancing the capillary-driven flow with a centrifugal force]]<br />
<br />
[[Two-dimensional nanometric confinement of entangled polymer melts]]<br />
<br />
[[Restructuring of Hydrophobic Surfaces Created by Surfactant Adsorption to Mica Surfaces]]<br />
<br />
[[The Phase Behavior of a Polymer-Fullerene Bulk Heterojunction System that Contains Bimolecular Crystals]]<br />
<br />
[[Concentration of Magnetic Beads Utilizing Light-Induced Electro-Osmosis Flow]]<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]<br />
<br />
[[Amphiphilic]]<br />
<br />
[[Centrifugal forces]<br />
<br />
[[Coulomb interaction]]<br />
<br />
[[Laplace pressure]]<br />
<br />
[[Photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23362Photonic crystals2011-12-09T05:49:17Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential. An example of photonic band gaps is shown in Figure 2.<br />
<br />
== Categories of photonic crystals ==<br />
<br />
Photonic crystals can be classified based on their dimensions of periodicity, i.e. in how many dimensions the nanostructures repeat themselves. 1D photonic crystals are often alternating sequence of layers with different dielectric constants, for instance [[Bragg mirrors]]. 2D photonic crystals usually consist of periodic holes or rods in a dielectric medium, and provide 2D photonic band gaps. 3D photonic crystals are the most difficult to fabricate, and proposed constructions of such 3D periodic nanostructures include spheres in a diamond lattice, Yablonovite, the woodpile crystal, inverse opals and stacks of two-dimensional crystals [2].<br />
<br />
== Applications ==<br />
<br />
Photonic crystals have vast application thanks to their distinct and engineerable optical properties [2]. For example, 1D photonic crystals can be used as [[omni-directional reflectors]] and low-loss [[optical filters]]. 2D photonic crystals can serve as logic devices such as [[optical switches]], and be fabricated into photonic crystal [[fibers]] and [[waveguides]]. The development of 3D photonic crystal is still far from mature due to the difficulty with fabrication, and may have great potential in areas such as optical computation etc [1].<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23361Photonic crystals2011-12-09T05:48:56Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential. An example of photonic band gaps is shown in Figure 2.<br />
<br />
== Categories of photonic crystals ==<br />
<br />
Photonic crystals can be classified based on their dimensions of periodicity, i.e. in how many dimensions the nanostructures repeat themselves. 1D photonic crystals are often alternating sequence of layers with different dielectric constants, for instance [[Bragg mirrors]]. 2D photonic crystals usually consist of periodic holes or rods in a dielectric medium, and provide 2D photonic band gaps. 3D photonic crystals are the most difficult to fabricate, and proposed constructions of such 3D periodic nanostructures include spheres in a diamond lattice, Yablonovite, the woodpile crystal, inverse opals and stacks of two-dimensional crystals [2].<br />
<br />
== Applications ==<br />
<br />
Photonic crystals have vast application thanks to their distinct and engineerable optical properties [2]. For example, 1D photonic crystals can be used as [[omni-directional reflectors]] and low-loss [[optical filters]]. 2D photonic crystals can serve as logic devices such as [[optical switches]], and be fabricated into photonic crystal [[fibers]] and [[waveguides]]. The development of 3D photonic crystal is still far from mature due to the difficulty with fabrication, and may find great potential in areas such as optical computation etc.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23360Photonic crystals2011-12-09T05:48:44Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential. An example of photonic band gaps is shown in Figure 2.<br />
<br />
== Categories of photonic crystals ==<br />
<br />
Photonic crystals can be classified based on their dimensions of periodicity, i.e. in how many dimensions the nanostructures repeat themselves. 1D photonic crystals are often alternating sequence of layers with different dielectric constants, for instance [[Bragg mirrors]]. 2D photonic crystals usually consist of periodic holes or rods in a dielectric medium, and provide 2D photonic band gaps. 3D photonic crystals are the most difficult to fabricate, and proposed constructions of such 3D periodic nanostructures include spheres in a diamond lattice, Yablonovite, the woodpile crystal, inverse opals and stacks of two-dimensional crystals [2].<br />
<br />
== Applications ==<br />
<br />
Photonic crystals have vast application thanks to their distinct and engineerable optical properties [2]. For example, 1D photonic crystals can be used as [[omni-directional reflectors]] and low-loss [[optical filters]]. 2D photonic crystals can serve as logic devices such as [[optical switches]], and be fabricated into photonic crystal [[fibers\\ and [[waveguides]]. The development of 3D photonic crystal is still far from mature due to the difficulty with fabrication, and may find great potential in areas such as optical computation etc.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23359Photonic crystals2011-12-09T05:48:35Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential. An example of photonic band gaps is shown in Figure 2.<br />
<br />
== Categories of photonic crystals ==<br />
<br />
Photonic crystals can be classified based on their dimensions of periodicity, i.e. in how many dimensions the nanostructures repeat themselves. 1D photonic crystals are often alternating sequence of layers with different dielectric constants, for instance [[Bragg mirrors]]. 2D photonic crystals usually consist of periodic holes or rods in a dielectric medium, and provide 2D photonic band gaps. 3D photonic crystals are the most difficult to fabricate, and proposed constructions of such 3D periodic nanostructures include spheres in a diamond lattice, Yablonovite, the woodpile crystal, inverse opals and stacks of two-dimensional crystals [2].<br />
<br />
== Applications ==<br />
<br />
Photonic crystals have vast application thanks to their distinct and engineerable optical properties [2]. For example, 1D photonic crystals can be used as [[omni-directional reflectors]] and low-loss [[optical filters]]. 2D photonic crystals can serve as logic devices such as [[optical switches\\, and be fabricated into photonic crystal [[fibers\\ and [[waveguides]]. The development of 3D photonic crystal is still far from mature due to the difficulty with fabrication, and may find great potential in areas such as optical computation etc.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23358Photonic crystals2011-12-09T05:47:48Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential. An example of photonic band gaps is shown in Figure 2.<br />
<br />
== Categories of photonic crystals ==<br />
<br />
Photonic crystals can be classified based on their dimensions of periodicity, i.e. in how many dimensions the nanostructures repeat themselves. 1D photonic crystals are often alternating sequence of layers with different dielectric constants, for instance [[Bragg mirrors]]. 2D photonic crystals usually consist of periodic holes or rods in a dielectric medium, and provide 2D photonic band gaps. 3D photonic crystals are the most difficult to fabricate, and proposed constructions of such 3D periodic nanostructures include spheres in a diamond lattice, Yablonovite, the woodpile crystal, inverse opals and stacks of two-dimensional crystals [2].<br />
<br />
== Applications ==<br />
<br />
Photonic crystals have vast application thanks to their distinct and engineerable optical properties [2]. For example, 1D photonic crystals can be used as omni-directional reflectors and low-loss optical filters. 2D photonic crystals can serve as logic devices such as optical switches, and be fabricated into photonic crystal fibers and waveguides. The development of 3D photonic crystal is still far from mature due to the difficulty with fabrication, and may find great potential in areas such as optical computation etc.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23357Photonic crystals2011-12-09T05:25:44Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals, taken from Ref [2].]]<br />
<br />
[[Image:PC2.png|200px|thumb|right|'''Fig.2 ''' A complete photonic band gap of a 1D photonic crystal, taken from Ref [2].]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=File:PC2.png&diff=23356File:PC2.png2011-12-09T05:25:10Z<p>Yjin: </p>
<hr />
<div></div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23355Photonic crystals2011-12-09T05:22:12Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:PC1.png|400px|thumb|right|'''Fig.1 ''' Illustrations of 1D, 2D and 3D photonic crystals.]]<br />
<br />
Photonic crystals are periodic dielectric [[nanostructures]] that can modulate the propagation of light. Usually a photonic crystal contains periodically repeating nano-sized areas of materials with sufficiently different [[dielectric constants]], as displayed in Figure 1. If the [[absorption of light]] by these components is small, light can be prevented from propagating in certain directions with certain frequencies due to the [[Bragg-like reflection]]. In other words, photonic band gaps are created in photonic crystals, similar to the [[energy gaps]] in the motion of electrons in solid state [[crystals]]. Here, the macroscopic media with distinct dielectric constants are analogous to the atoms or molecules in a crystal, and the periodic [[refraction index]] experienced by the photons is analogous to the periodic Coulomb potential.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=File:PC1.png&diff=23354File:PC1.png2011-12-09T05:21:09Z<p>Yjin: </p>
<hr />
<div></div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Photonic_crystals&diff=23353Photonic crystals2011-12-09T04:57:46Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== References ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Photonic_crystals Wikipedia of photonic crystals]<br />
<br />
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, & R. D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2008.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Encoding complex wettability patterns in chemically functionalized 3D photonic crystals]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Laplace_pressure&diff=23352Laplace pressure2011-12-09T04:55:13Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
The Laplace pressure is the pressure difference across a curved surface or interface [2]. This pressure jump arises from [[surface tension]] or [[interfacial tension]], whose presence tends to compress the curved surface or interface. At equilibrium, this trend is balanced by an extra pressure at the concave side. The Laplace pressure is given as<br />
<br />
<math>P_L=\gamma(\frac{1}{R_1}+\frac{1}{R_2})</math>,<br />
<br />
where <math>R_1</math> and <math>R_2</math> are the orthogonal (principle) radii of the curvature, and <math>\gamma</math> is the surface tension or interfacial tension. For a spherical droplet or bubble with a radius of <math>R</math>, the formula reduces to<br />
<br />
<math>P_L=2\frac{\gamma}{R}</math>.<br />
<br />
The Laplace pressure is usually insignificant for macroscopic droplets or bubbles with a diameter of 10 cm or larger. However, this [[capillary effect]] is especially important for small bubbles. For instance, an air bubble in water whose diameter is 1 μm can have an extra pressure of 2.9 atm inside. This explains the extra energy required to generate [[emulsions]] such as droplets in [[droplet microfluidics]] etc.<br />
<br />
== References ==<br />
<br />
[1] [[High-throughput injection with microfluidics using picoinjectors]]<br />
<br />
[2] J. N. Israelachvili, "Intermolecular and Surface Forces", Academic Press, 2011.<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A non equilibrium mechanism for nanobubble stabilization]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Laplace_pressure&diff=23351Laplace pressure2011-12-09T04:53:41Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
The Laplace pressure is the pressure difference across a curved surface or interface [2]. This pressure jump arises from [[surface tension]] or [[interfacial tension]], whose presence tends to compress the curved surface or interface. At equilibrium, this trend is balanced by an extra pressure at the concave side. The Laplace pressure is given as<br />
<br />
<math>P_L=\gamma(\frac{1}{R_1}+\frac{1}{R_2})</math>,<br />
<br />
where <math>R_1</math> and <math>R_2</math> are the orthogonal (principle) radii of the curvature, and <math>\gamma</math> is the surface tension or interfacial tension. For a spherical droplet or bubble with a radius of <math>R</math>, the formula reduces to<br />
<br />
<math>P_L=2\frac{\gamma}{R}</math>.<br />
<br />
The Laplace pressure is usually insignificant for macroscopic droplets or bubbles with a diameter of 10 cm or larger. However, this [[capillary effect]] is especially important for small bubbles. For instance, an air bubble in water whose diameter is 1 μm can have an extra pressure of 2.9 atm inside. This explains the extra energy required to generate [[emulsions]] such as droplets in [[droplet microfluidics]] etc.<br />
<br />
==References==<br />
<br />
[1] [[High-throughput injection with microfluidics using picoinjectors]]<br />
<br />
[2] J. N. Israelachvili, "Intermolecular and Surface Forces", Academic Press, 2011.<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A non equilibrium mechanism for nanobubble stabilization]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Laplace_pressure&diff=23348Laplace pressure2011-12-09T04:44:36Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
The Laplace pressure is the pressure difference across a curved surface or interface. This pressure jump arises from [[surface tension]] or [[interfacial tension]], whose presence tends to compress the curved surface or interface. At equilibrium, this trend is balanced by an extra pressure at the concave side. The Laplace pressure is given as<br />
<br />
<math>P_L=\gamma(\frac{1}{R_1}+\frac{1}{R_2})</math>,<br />
<br />
where <math>R_1</math> and <math>R_2</math> are the orthogonal (principle) radii of the curvature, and <math>\gamma</math> is the surface tension or interfacial tension. For a spherical droplet or bubble with a radius of <math>R</math>, the formula reduces to<br />
<br />
<math>P_L=2\frac{\gamma}{R}</math>.<br />
<br />
The Laplace pressure is usually insignificant for droplets of macroscopic bubbles with a diameter of 10 cm or larger. However, this [[capillary effect]] is especially important for small bubbles. For instance, an air bubble in water whose diameter is 1 μm can have an extra pressure of 2.9 atm inside. This explains the extra energy required to generate [[emulsions]] such as droplets in [[droplet microfluidics]] etc.<br />
<br />
==References==<br />
<br />
[1] [[High-throughput injection with microfluidics using picoinjectors]]<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A non equilibrium mechanism for nanobubble stabilization]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Laplace_pressure&diff=23346Laplace pressure2011-12-09T04:31:53Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
The Laplace pressure is the pressure difference across a curved surface or interface. This pressure jump arises from [[surface tension]] or [[interfacial tension]], whose presence tends to compress the curved surface or interface. At equilibrium, this trend is balanced by an extra pressure at the concave side. The Laplace pressure is given as<br />
<br />
<math>P_L=\gamma(\frac{1}{R_1}+\frac{1}{R_2})</math>,<br />
<br />
where <math>R_1</math> and <math>R_2</math> are the orthogonal (principle) radii of the curvature, and <math>\gamma</math> is the surface tension or interfacial tension. For a spherical droplet or bubble with a radius of <math>R</math>, the formula reduces to<br />
<br />
<math>P_L=2\frac{\gamma}{R}</math>.<br />
<br />
==References==<br />
<br />
[1] [[High-throughput injection with microfluidics using picoinjectors]]<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A non equilibrium mechanism for nanobubble stabilization]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Laplace_pressure&diff=23344Laplace pressure2011-12-09T04:28:09Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
The Laplace pressure is the pressure difference across a curved surface or interface. This pressure jump arises from [[surface tension]] or [[interfacial tension]], whose presence tends to compress the curved surface or interface. At equilibrium, this trend is balanced by an extra pressure at the concave side. The Laplace pressure is given as<br />
<br />
<math>P_L=\gamma(\frac{1}{R_1}+\frac{1}{R_2})</math>,<br />
<br />
where <br />
<br />
<br />
<br />
<br />
<br />
==References==<br />
<br />
[1] [[High-throughput injection with microfluidics using picoinjectors]]<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A non equilibrium mechanism for nanobubble stabilization]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23341Amphiphilic2011-12-09T03:25:04Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|left|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps, detergants and [[surfactants]] [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2. More importantly, due to their unique properties, amphiphilic molecules as surfactants can be absorbed to water-oil interfaces, reducing the [[interfacial energy]] and facilitating the formation of [[emulsions]] such as droplets. This role of amphiphilic molecules has been vastly utilized in [[droplet microfluidics]].<br />
<br />
== Reference ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Amphiphile Wikipedia of amphiphile]<br />
<br />
[2] Witten, T. A. & Pincus, P., "Structured Fluids", Oxford University Press, 2004<br />
<br />
[3] [http://en.wikipedia.org/wiki/Surfactant Wikipedia of surfactant]<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23340Amphiphilic2011-12-09T03:24:30Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|left|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps, detergants and [[surfactants]] [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2. More importantly, due to their unique properties, amphiphilic molecules as surfactants can be absorbed to water-oil interfaces, reducing the interfacial energy and facilitating the formation of emulsions such as droplets. This role of amphiphilic molecules has been vastly utilized in [[droplet microfluidics]].<br />
<br />
== Reference ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Amphiphile Wikipedia of amphiphile]<br />
<br />
[2] Witten, T. A. & Pincus, P., "Structured Fluids", Oxford University Press, 2004<br />
<br />
[3] [http://en.wikipedia.org/wiki/Surfactant Wikipedia of surfactant]<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23339Amphiphilic2011-12-09T03:24:12Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|right|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps, detergants and [[surfactants]] [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2. More importantly, due to their unique properties, amphiphilic molecules as surfactants can be absorbed to water-oil interfaces, reducing the interfacial energy and facilitating the formation of emulsions such as droplets. This role of amphiphilic molecules has been vastly utilized in [[droplet microfluidics]].<br />
<br />
== Reference ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Amphiphile Wikipedia of amphiphile]<br />
<br />
[2] Witten, T. A. & Pincus, P., "Structured Fluids", Oxford University Press, 2004<br />
<br />
[3] [http://en.wikipedia.org/wiki/Surfactant Wikipedia of surfactant]<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23338Amphiphilic2011-12-09T03:23:14Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
[[Image:AM1.jpg|300px|thumb|right|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps, detergants and [[surfactants]] [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2. More importantly, due to their unique properties, amphiphilic molecules as surfactants can be absorbed to water-oil interfaces, reducing the interfacial energy and facilitating the formation of emulsions such as droplets. This role of amphiphilic molecules has been vastly utilized in [[droplet microfluidics]].<br />
<br />
== Reference ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Amphiphile Wikipedia of amphiphile]<br />
<br />
[2] Witten, T. A., "Structured fluids<br />
<br />
[3] [http://en.wikipedia.org/wiki/Surfactant Wikipedia of surfactant]<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23336Amphiphilic2011-12-09T03:22:03Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|left|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps, detergants and [[surfactants]] [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2. More importantly, due to their unique properties, amphiphilic molecules as surfactants can be absorbed to water-oil interfaces, reducing the interfacial energy and facilitating the formation of emulsions such as droplets. This role of amphiphilic molecules has been vastly utilized in [[droplet microfluidics]].<br />
<br />
== Reference ==<br />
<br />
[1] [http://en.wikipedia.org/wiki/Amphiphile Wikipedia of amphiphile]<br />
<br />
[2] Witten, T. A., "Structured fluids<br />
<br />
[3] [http://en.wikipedia.org/wiki/Surfactant Wikipedia of surfactant]<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23327Amphiphilic2011-12-08T22:51:13Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|left|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps and surfactants [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23326Amphiphilic2011-12-08T22:51:00Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|right|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [2]]]<br />
<br />
[[Image:AM2.jpg|200px|thumb|right|'''Fig.2 ''' Structure of a spherical micelle taken from Ref. [3]]]<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures [1]. The structure of a typical amphiphilic molecule SDS is shown in Figure 1.<br />
<br />
The most common examples of amphiphilic chemicals are soaps and surfactants [1,2]. Their special structures lead to relatively high solubility in both [[polar solvents]] like water and a large variety of [[nonpolar solvents]]. Amphiphilic molecules may form [[aggregates]] or [[micelles]] in water. An example of a spherical micelle is presented in Figure 2.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=File:AM2.jpg&diff=23325File:AM2.jpg2011-12-08T22:50:08Z<p>Yjin: </p>
<hr />
<div></div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23324Coulomb interaction2011-12-08T22:40:30Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law [1]:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Applications of Coulomb interaction ==<br />
<br />
Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of [[electrical double layers]] and the stabilization of [[colloids]] [2].<br />
<br />
== References ==<br />
<br />
[1] Jackson, J. D., "Classical Electrodynamics", John Wiley & Sons, 1999.<br />
<br />
[2] Lecture notes of AP225.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Centrifugal_forces&diff=23323Centrifugal forces2011-12-08T22:40:04Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Centrifugal forces are usually referred to as [[fictitious forces]] ([[inertial forces]]) that arise in a [[rotating frame of reference]]. A centrifugal force represents the inertia of a rotating body, and is directed away from the center of axis of rotation. This concept can be generalized in [[Lagrangian mechanics]] when [[generalized coordinates]] are in effect. At times centrifugal forces may also denote the [[reaction forces]] in response to [[centripetal forces]].<br />
<br />
== General formula of centrifugal forces as inertial forces ==<br />
<br />
When the motion of a body is studied in a [[non-inertial frame of reference]], fictitious forces are introduced for convenience [1]. The connection between velocity in an [[inertial frame of reference]] and that in a frame rotating at an angular velocity of <math>\mathbf\Omega</math> is given as<br />
<br />
<math>\mathbf v=\mathbf v'+\mathbf\Omega\times\mathbf r</math>,<br />
<br />
where <math>\mathbf r'</math> denotes the displacement in the rotating fram, <math>\mathbf v</math> the absolute velocity (in an inertial frame of reference) and <math>\mathbf v'</math> the velocity in the rotating frame. Similarly, the relation of acceleration is given as<br />
<br />
<math>\mathbf a=\mathbf a'+\dot{\mathbf\Omega}\times\mathbf r'+2\mathbf\Omega\times\mathbf v'+\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math>.<br />
<br />
Applying the [[Second Law of Newtonian mechanics]] we have<br />
<br />
<math>\mathbf F=m\mathbf a=m\mathbf a'+m\dot{\mathbf\Omega}\times\mathbf r'+2m\mathbf\Omega\times\mathbf v'+m\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math>,<br />
<br />
i.e.<br />
<br />
<math>m\mathbf a'=\mathbf F-m\dot{\mathbf\Omega}\times\mathbf r'-2m\mathbf\Omega\times\mathbf v'-m\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math>.<br />
<br />
Hence from the perspective of this rotating frame of reference, the terms other than <math>\mathbf F</math> on the right hand side of the equation are fictitious forces. Specifically <math>-m\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math> is called the centrifugal force, since it points outward perpendicular to <math>\mathbf\Omega</math>. Therefore, centrifugal forces are proportional to the distance to the axis of rotation as well as the square of the angular velocity of the rotating frame.<br />
<br />
== Centrifugal forces as reactive forces ==<br />
<br />
In some contexts, centrifugal forces refer to reactive forces [2]. The motion of a rotating body is maintained by a [[centripetal force]] provided by another object. According to the [[Third Law of Newtonian mechanics]], the rotating body exerts a reactive force on that object, referred to as a centrifugal force.<br />
<br />
== Applications ==<br />
<br />
Many devices make use of centrifugal forces, such as centrifuges and centrifugal pumps, which have found numerous applications in the industry and academia. Centrifugal forces are also an important factor in engineering designs for railways and satellites etc. Recently, in space stations centrifugal forces are used to balance gravity to approximate zero-gravity environments.<br />
<br />
== References ==<br />
<br />
[1] Gregory, R. D., "Classical Mechanics", Cambridge University Press, 2006.<br />
<br />
[2] [http://en.wikipedia.org/wiki/Centrifugal_force Wikipedia on centrifugal forces]<br />
<br />
== Keyword in references: ==<br />
<br />
[[Paper on a disc: balancing the capillary-driven flow with a centrifugal force]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23322Amphiphilic2011-12-08T22:39:03Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
[[Image:AM1.jpg|300px|thumb|right|'''Fig.1 ''' Structure and model of an amphiphilic molecule sodium dodecyl sulfate (SDS) taken from Ref. [1]]]<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures. The structure of a typical amphiphilic molecule SDS is shown in Figure 1. This special structure leads to relatively high solubility in both polar solvents like water and a large variety of nonpolar solvents.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=File:AM1.jpg&diff=23321File:AM1.jpg2011-12-08T22:37:58Z<p>Yjin: </p>
<hr />
<div></div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23320Amphiphilic2011-12-08T22:36:42Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing a large number of polar, non-charged hydroxyl groups etc. can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures. The structure of a typical amphiphilic molecule SDS is shown in Figure 1. This special structure leads to relatively high solubility in both polar solvents like water and a large variety of nonpolar solvents.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23319Amphiphilic2011-12-08T22:32:52Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -CO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges. Sometimes a structure containing polar hydroxyl groups can also serve as the hydrophilic part. The lipophilic part is usually a long chain of nonpolar hydrocarbon structures. This special structure leads to relatively high solubility in both polar solvents like water and a large variety of nonpolar solvents.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23318Amphiphilic2011-12-08T22:22:15Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is used to describe the properties of a certain group of chemicals. A chemical substance being amphiphilic possesses both [[hydrophilic]] and [[lipophilic]] functional groups. As a result, amphiphilic compounds show good affinity for both water and oil, and have relatively high solubility in polar solvents like water and some non-polar solvents.<br />
<br />
== Structure and examples of amphiphilic molecules ==<br />
<br />
In amphiphilic molecules, the hydrophilic group is usually charged and highly polar. Examples include carboxylates -RCO<sub>2</sub><sup>-</sup> and sulfonates -SO<sub>3</sub><sup>-</sup> with negative charges and amines -NH<sub>3</sub><sup>+</sup> with positive charges.<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Amphiphilic&diff=23316Amphiphilic2011-12-08T22:04:54Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
"Amphiphilic" is a term used to describe the properties of chemicals. A chemical substance being amphiphilic <br />
<br />
<br />
<br />
See also:<br />
<br />
[[Etymology and organization of surfactants#Etymology|Amphiphilic]] in [[Surfactants]] from [[Main Page#Lectures for AP225|Lectures for AP225]].<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Amphiphilic Crescent-Moon-Shaped Microparticles Formed by Selective Adsorption of Colloids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23315Coulomb interaction2011-12-08T22:00:49Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Applications of Coulomb interaction ==<br />
<br />
Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of [[electrical double layers]] and the stabilization of [[colloids]].<br />
<br />
== References ==<br />
<br />
[1] Jackson, J. D., "Classical Electrodynamics", John Wiley & Sons, 1999.<br />
<br />
[2] Lecture notes of AP225.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23314Coulomb interaction2011-12-08T22:00:22Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Applications of Coulomb interaction ==<br />
<br />
Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of [[electrical double layers]] and the stabilization of [[colloids]].<br />
<br />
== References ==<br />
<br />
[1] Jackson, J. D., "Classical Electrodynamics", John Wiley & Sons, 1999.<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23313Coulomb interaction2011-12-08T21:57:31Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Applications of Coulomb interaction ==<br />
<br />
Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of [[electrical double layers]] and the stabilization of [[colloids]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23312Coulomb interaction2011-12-08T21:57:10Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Applications of Coulomb interaction ==<br />
<br />
Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of [[electrical double layer]] and the stabilization of [[colloids]].<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23308Coulomb interaction2011-12-08T21:45:57Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23307Coulomb interaction2011-12-08T21:44:31Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}</math>.<br />
<br />
Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is<br />
<br />
<math>\U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2</math>,<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23306Coulomb interaction2011-12-08T21:41:48Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The force experienced by another point charge <math>q_2</math> due to <math>q_1</math> is given by<br />
<br />
<math>\mathbf F_{21}=-q_2\nabla U</math>.<br />
<br />
The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}</math>.<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23304Coulomb interaction2011-12-08T21:37:47Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic properties and formulae of Coulomb interaction ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
Coulomb forces are conservative, and as a result Coulomb interaction can be described using the [[Coulomb potential]]. The potential created by a point charge <math>q_1</math> is<br />
<br />
<math>U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}</math>,<br />
<br />
where <math>r</math> is the distance from <math>q_1</math> charge. The interaction potential between two point charges is given as<br />
<br />
<math>U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}</math>.<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23303Coulomb interaction2011-12-08T21:32:05Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge at position <math>\mathbf r</math> exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>,<br />
<br />
where <math>\rho(\mathbf r')</math> is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is<br />
<br />
<math>\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2</math>.<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23302Coulomb interaction2011-12-08T21:29:49Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force experience by <math>q_1</math> (solely) due to the presence of <math>q_2</math>, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
Coulomb interaction obeys the [[principle of linear superstition]], and therefore the force on a point charge exerted by a object with continuous charge distribution is give as<br />
<br />
<math>\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'</math>.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23299Coulomb interaction2011-12-08T21:22:51Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> at position <math>\mathbf r_1</math> and <math>q_2</math> at position <math>\mathbf r_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23298Coulomb interaction2011-12-08T21:21:10Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of <math>8.85\times10^{-12}</math> approximately.<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23297Coulomb interaction2011-12-08T21:20:58Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]], which has a value of </math>8.85\times10^{-12}</math> approximately.<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23296Coulomb interaction2011-12-08T21:20:20Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]].<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23295Coulomb interaction2011-12-08T21:20:11Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]].<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23294Coulomb interaction2011-12-08T21:19:59Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\epsilon_0}{\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]].<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjinhttp://soft-matter.seas.harvard.edu/index.php?title=Coulomb_interaction&diff=23293Coulomb interaction2011-12-08T21:19:35Z<p>Yjin: </p>
<hr />
<div>Written by [[Yuhang Jin]], AP225 2011 Fall.<br />
<br />
== Introduction ==<br />
<br />
Coulomb interaction is the electrostatic interactions between electric charges, and follows the [[Coulomb's law]], which is a basis of [[classical electrodynamics]]. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of [[quantum field theory]].<br />
<br />
== Basic formulae ==<br />
<br />
The electrostatic interaction of two point charges <math>q_1</math> and <math>q_2</math> is described by the Coulomb's law:<br />
<br />
<math>\mathbf F=\frac{1}{4\pi\epsilon_0}{\frac{q_1q_2}{r^2}</math>,<br />
<br />
where <math>\mathbf F</math> is the electrostatic force, <math>r</math> denotes the distance between the two charges, and <math>\epsilon_0</math> is the [[vacuum permittivity]].<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Photonic Properties of Strongly Correlated Colloidal Liquids]]</div>Yjin