Coefficients of potential

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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 φ_i is the electrical potential on a conductor surface S_i and Q_i is the surface charge on conductor i. The coefficients of potential are the coefficients p_ij.

[p_ = = \left( \right)_, Q_,...,Q_n},]

or more formally

[p_ = \frac\int_\frac}.]

Note that:

p_ij = p_ji, by symmetry, and
p_ij is not dependent on the charge,

The physical content of the symmetry is as follow:

if a charge Q on conductor j brings conductor i to a potential φ, then the same charge placed on i would bring j to the same potential φ.

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 P is [\phi_P = \sum_^\frac\int_\frac}].

Given the electrical potential on a conductor surface S_i (the equipotential surface or the point P chosen on surface i) contained in a system of conductors j = 1, 2, ..., n:

[\phi_i = \sum_^\frac\int_\frac} \mbox,]

where R_ji = |r_i - r_j|, i.e. the distance from the area-element da_j to a particular point r_i on conductor i. σ_j is not, in general, uniformly distributed across the surface. Let us introduce the factor f_j that describes how the actual charge density differs from the average and itself on a position on the surface of the j-th conductor:

[\frac = f_j,]

[\sigma_j = <\sigma_j>f_j = \fracf_j.]

Then,

[\phi_i = \sum_^n\frac\int_\frac}]

can be written in the form

[\phi_i=\sum_^n p_Q_j \mbox, ]

i.e.

[p_ = \frac\int_\frac}.]

Example

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: Q = Q₁ = -Q₂. 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 + p_ - 2p_}.]

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 c_ii is called the coefficients of capacitance and the c_ij with i ≠ j is called the coefficients of induction.

The capacitance of this system can be expressed as

[C = \fracc_ - c_^2} + c_ + 2c_}]

(the system of conductors can be shown to have similar symmetry c_ij = c_ij.)

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