Coefficients Of Potential

In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric:
:$\begin \phi_1 = p_Q_1 + \cdots + p_Q_n \\ \phi_2 = p_Q_1 + \cdots + p_Q_n \\ \vdots \\ \phi_n = p_Q_1 + \cdots + p_Q_n \end.$

where is the surface charge on conductor . The coefficients of potential are the coefficients . should be correctly read as the potential on the -th conductor, and hence "$p_$" is the due to charge 1 on conductor 2.
:$p_ = = \left( \right)_.$

Note that:
# , by symmetry, and
# is not dependent on the charge.

The physical content of the symmetry is as follows:
: if a charge on conductor brings conductor to a potential , then the same charge placed on would bring to the same potential .

In general, the coefficients is used when describing system of conductors, such as in the capacitor.

Theory

System of conductors. The electrostatic potential at point is $\phi_P = \sum_^\frac\int_\frac$.

Given the electrical potential on a conductor surface (the equipotential surface or the point chosen on surface ) contained in a system of conductors :
:$\phi_i = \sum_^\frac\int_\frac \mbox,$

where , i.e. the distance from the area-element to a particular point on conductor . is not, in general, uniformly distributed across the surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor:
:$\frac = f_j,$
or
: $\sigma_j = \langle\sigma_j\rangle f_j = \fracf_j.$
Then,
:$\phi_i = \sum_^n\frac\int_\frac.$
It can be shown that $\int_\frac$ is independent of the distribution $\sigma_j$. Hence, with
:$p_ = \frac\int_\frac,$
we have
:$\phi_i=\sum_^n p_Q_j \mbox.$

Example

In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system.

For a two-conductor system, the system of linear equations is
:$\begin \phi_1 = p_Q_1 + p_Q_2 \\ \phi_2 = p_Q_1 + p_Q_2 \end.$

On a capacitor, the charge on the two conductors is equal and opposite: . Therefore,
:$\begin \phi_1 = (p_ - p_)Q \\ \phi_2 = (p_ - p_)Q \end,$
and
:$\Delta\phi = \phi_1 - \phi_2 = (p_ + p_ - p_ - p_)Q.$
Hence,
: $C = \frac.$

Related coefficients

Note that the array of linear equations
:$\phi_i = \sum_^n p_Q_j \mbox$
can be inverted to
:$Q_i = \sum_^n c_\phi_j \mbox$
where the with are called the coefficients of capacity and the with are called the coefficients of electrostatic induction.

For a system of two spherical conductors held at the same potential,
:$Q_a=(c_+c_)V , \qquad Q_b=(c_+c_)V$

$Q =Q_a+Q_b =(c_+2c_+c_)V$

If the two conductors carry equal and opposite charges,
:$\phi_1=\frac , \qquad \quad \phi_2=\frac$
$\quad C =\frac= \frac$

The system of conductors can be shown to have similar symmetry .

References

{{Reflist
* James Clerk Maxwell (1873) A Treatise on Electricity and Magnetism, § 86, page 89.

Electrostatics