# Electrostatics

## What is electrostatics?

Electrostatics is the branch of science that deals with the phenomena arising from what seems to be stationary electric charges.

Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον (electron), was the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, the electrostatic force between e.g an electron and a proton, that together make up a hydrogen atom, is about 40 orders of magnitude stronger than the gravitational force acting between them.

Electrostatic phenomena include examples as simple as the attraction of plastic wrap to your hand after you remove it from a package, to the apparently spontaneous explosion of grain silos, to damage of electronic components during manufacturing, to the operation of photocopiers. Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer to or from the highly resistive surface are more or less trapped there for a long enough time for their effects to be observed. These charges then remain on the object until they either bleed off to ground or are quickly neutralized by a discharge: e.g., the familiar phenomenon of a static 'shock' is caused by the neutralization of charge built up in the body from contact with nonconductive surfaces.

## Introduction

The following examples are taken from an excellent book on electrostatics:

J.M. Crowley, Fundamentals of applied electrostatics, Wiley & Sons, 1986

## Volts, fields, charges

 Crowley Crowley Crowley

### Biological Cells

Comparison of action potentials (APs) from a representative cross-section of animals-Theodore Holmes Bullock, 1965 "Structure and Function in the Nervous Systems of Invertebrates."
Animal Cell type Resting potential (mV) AP increase (mV) AP duration (ms) Conduction speed (m/s)
Squid (Loligo) Giant axon −60 120 0.75 35
Earthworm (Lumbricus) Median giant fiber −70 100 1.0 30
Cockroach (Periplaneta) Giant fiber −70 80–104 0.4 10
Frog (Rana) sciatic nerve axon −60 to −80 110–130 1.0 7–30
Cat (Felis) Spinal motor neuron −55 to −80 80–110 1–1.5 30–120

Charged surfaces are very important in biological systems since all nerve cells use gradients of ions in cellular communication.

• Side Note*** Connection to previous topic: A liquid crystal bilayer is very very important in the function of nerve cells. Cell membranes are liquid crystal bilayers and because of their structure and local charge they are practically impervious to ions! Without this the body would not beable to build up gradients in ions to activate communication throughout the body.

#### Cable Theory View of Neuron Communication

Electrical response in axons of nerve cells can be mathematically described using cable theory generated by Lord Kelvin in 1855 for modeling the transatlantic telegraph cable. An axon can be approximated as an "electrically passive, perfectly cylindrical transmission cable" described by the following differential equation:

$\tau \frac{\partial V}{\partial t} = \lambda^{2} \frac{\partial^{2} V}{\partial x^{2}} - V$

where V(x, t) is the voltage across the membrane at a time t and a position x along the length of the neuron, and where λ and τ are the characteristic length and time scales on which those voltages decay in response to a stimulus.

Referring to the circuit diagram above, these scales can be determined from the resistances and capacitances per unit length

$\tau =\ r_{m} c_{m}$

$\lambda = \sqrt \frac{r_m}{r_l}$

"These time- and length-scales can be used to understand the dependence of the conduction velocity on the diameter of the neuron in unmyelinated fibers. For example, the time-scale τ increases with both the membrane resistance rm and capacitance cm. As the capacitance increases, more charge must be transferred to produce a given transmembrane voltage as the resistance increases, less charge is transferred per unit time, making the equilibration slower. Similarly, if the internal resistance per unit length ri is lower in one axon than in another (e.g., because the radius of the former is larger), the spatial decay length λ becomes longer and the conduction velocity of an action potential should increase. If the transmembrane resistance rm is increased, that lowers the average "leakage" current across the membrane, likewise causing λ to become longer, increasing the conduction velocity."

For many more references see original compilation here.[1]

## Voltages and electric fields

 Electric force is conservative. (Even in Massachusetts.) $\oint{qE\cdot dr}=0=\nabla \times E$ Crowley Fig. 1.1.1 $\oint{E\cdot dr}=\int\limits_{-\text{terninal}}^{+\text{terminal}}{E\cdot dr+}\int\limits_{\text{wires}}^{{}}{E\cdot dr+}\int\limits_{\text{gap}}^{{}}{E\cdot dr}=0$ $v=\int\limits_{\text{+ terninal}}^{\text{- terminal}}{E\cdot dr}=Ed$

## Charges and electric fields

 Gauss’s law: $\oint\limits_{S}{D\cdot dA}=\oint\limits_{S}{\varepsilon E\cdot dA}=q$ D = displacement dA = area vector S = surface $\varepsilon$ = dielectric constant E = Electric field q = charge Another form is: \begin{align}  & \nabla \cdot D=\rho \\ & \nabla \cdot E=\frac{\rho }{\varepsilon } \\  \end{align}\,\! (when the dielectric constant is constant!) Crowley Fig. 1.2.1 $D=\frac{q}{4\pi r^{2}}\,\!$ Crowley Fig. 1.2.2 $\left( D_{a} \right)_{norm}-\left( D_{b} \right)_{norm}=\frac{q}{A}\equiv \rho _{s}\,\!$