Conductors in Electrostatic Fields

Electric Field of Conducting Sphere

Systems of Conductors

We have seen how to calculate the potential due to a system of point charges. If we have a system of n conductors, the potential of any conductor depends linearly on the charges of all of the conductors.

We may write then:

f₁ = p₁₁ Q₁ + p₁₂ Q₂ + ? + p_1nQ_n

f₂ = p₂₁ Q₁ + p₂₂ Q₂ + ? + p_2nQ_n

_?

f_n = p_n1 Q₁ + p_n2 Q₂ + ? + p_nnQ_n

The set of coefficient p_ij is called coeffcients of potential and in general their total number is n².

It is easy to show that p_ij = p_ji

These coefficients represent purely geometric relationships and are in principle measurable.

Example: Isolated conducting sphere of radius a.

We know that at the surface:

f = (1/ 4pe_o) Q/a

From the previous consideration, we also know that

f₁ = p₁₁ Q₁

thus we see here that

p₁₁ = (1/ 4pe_oa)

Capacitance

The capacitance of a single conductor is defined as

C = Q/f = 1 / p₁₁

Thus, for the sphere we see that

C = 4pe_oa

When we consider a system with only two conductors, then:

f₁ = p₁₁ Q₁ + p₁₂ Q₂

f₂ = p₂₁ Q₁ + p₂₂ Q₂

where  p₁₂ = p₂₁

If we now take our general definition of a capacitor as two conductors with equal and opposite charges Q and ? Q:

The difference in potential between the two is:

Df = f₁ - f₂ = (p₁₁ - p₁₂ -  p₂₁  + p₂₂  ) Q

and

C = Q /Df

Thus , in general, the capacitance can be expressed in terms of the coefficients of potential as:

C = 1 / (p₁₁ + p₂₂ ? 2p₁₂)

 

 

Spherical Capacitor       The capacitance for spherical or cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. By applying Gauss' law to an charged conducting sphere, the electric field outside it is found to be

 

The voltage between the spheres can be found by integrating the electric field along a radial line:

From the definition of capacitance, the capacitance is

Note: we could calculate this also using the concept of coefficients of potential:

If we call the inner conductor 1 and the outer conductor 2, and the outermost radius c, and we first assume that Q₁ = 0 and Q₂ not zero, then

f₁ =  p₁₂ Q₂

f₂ = p₂₂ Q₂

Since there is no charge in the inner conductor then f₂ = (1/ 4pe_o) Q₂/c

and p₂₂ = 1/ 4pe_oc. Also, f₁ = f₂ and this implies that p₁₂ =  p₂₂

In order to find p₁₁, we now assume that Q₁ is not zero and Q₂ = 0.

We find f₁ - f₂ by calculating the integral of E along a path that goes from radius a to radius b and we obtain:

f₁ - f₂ =  Q₁/4pe_o (1/a -1/b)

The left hand side can also be written as (p₁₁ ?p₂₁) Q₁

Thus,

 p₁₁ = p₂₂ + 1/4pe_o (1/a ? 1/b) = /4pe_o (1/a ? 1/b + 1/c); and because p₁₂ = p₂₂

 

 C = 1/(p₁₁-p₂₂) = 4pe_o [(1/a)-(1/b)] = 4pe_o ab /(b ? a)

as before.

 

In most cases this may not be the most convenient way to proceed. What is generally done is to find the potential difference from a different and prior solution to the problem and use the definition of C = Q/Df

 

Cylindrical Capacitor The capacitance for cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. By applying Gauss' law to an infinite cylinder, the electric field outside a charged cylinder is found to be

The voltage between the cylinders can be found by integrating the electric field along a radial line:

From the definition of capacitance, the capacitance per unit length is defined as

So, capacitance is typified by a parallel plate arrangement and is defined in terms of charge storage: where Q = magnitude of charge stored on each plate. V = voltage applied to the plates.

Parallel Plate Capacitor

The capacitance of flat, parallel metallic plates of area A and separation d is given by the expression above where:

= permittivity of space and

 

k = relative permittivity of the dielectric material between the plates.

k=1 for free space, k>1 for all media, approximately =1 for air. The Farad, F, is the SI unit for capacitance, and from the definition of capacitance is seen to be equal to a Coulomb/Volt.