`:top

In `F33f`_`[electrostatics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Electrostatics]`_`f, the `!`F33f`_`[coefficients`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Coefficients]`_`f of potential`! determine the relationship between the `F33f`_`[charge`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Electric_charge]`_`f and `F33f`_`[electrostatic potential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Electrostatic_potential]`_`f (`F33f`_`[electrical potential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Electrical_potential]`_`f), which is purely geometric:

ϕ ϕ 1 = p 11 Q 1 + ⋯ ⋯ + p 1 n Q n ϕ ϕ 2 = p 21 Q 1 + ⋯ ⋯ + p 2 n Q n ⋮ ⋮ ϕ ϕ n = p n 1 Q 1 + ⋯ ⋯ + p n n Q n . {\\displaystyle {\\begin{matrix}\\phi _{1}=p_{11}Q_{1}+\\cdots +p_{1n}Q_{n}\\\\\\phi _{2}=p_{21}Q_{1}+\\cdots +p_{2n}Q_{n}\\\\\\vdots \\\\\\phi _{n}=p_{n1}Q_{1}+\\cdots +p_{nn}Q_{n}\\end{matrix}}.}

where `*Q`*i is the `F33f`_`[surface charge`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Surface_charge]`_`f on conductor i. The coefficients of potential are the coefficients `*p`*ij. φi should be correctly read as the potential on the i-th conductor, and hence " p 21 {\\displaystyle p_{21}} " is the `*p`* due to charge 1 on conductor 2.

p i j = ∂ ∂ ϕ ϕ i ∂ ∂ Q j = ( ∂ ∂ ϕ ϕ i ∂ ∂ Q j ) Q 1 , . . . , Q j − − 1 , Q j + 1 , . . . , Q n . {\\displaystyle p_{ij}={\\partial \\phi _{i} \\over \\partial Q_{j}}=\\left({\\partial \\phi _{i} \\over \\partial Q_{j}}\\right)_{Q_{1},...,Q_{j-1},Q_{j+1},...,Q_{n}}.}

Note that:

1. `*p`*ij = `*p`*ji, by symmetry, and

2. `*p`*ij is not dependent on the charge.

The physical content of the symmetry is as follows:

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 `F33f`_`[capacitor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Capacitor]`_`f.

>>Contents

• `F0af`_`[Theory`#theory]`_`f

• `F0af`_`[Example`#example]`_`f

• `F0af`_`[Related coefficients`#related-coefficients]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Theory

System of conductors. The electrostatic potential at point `*P`* is ϕ ϕ P = ∑ ∑ j = 1 n 1 4 π π ϵ ϵ 0 ∫ ∫ S j σ σ j d a j R j {\\displaystyle \\phi _{P}=\\sum _{j=1}^{n}{\\frac {1}{4\\pi \\epsilon _{0}}}\\int _{S_{j}}{\\frac {\\sigma _{j}da_{j}}{R_{j}}}} .

Given the electrical potential on a conductor surface `*S`*i (the `F33f`_`[equipotential surface`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equipotential_surface]`_`f or the point `*P`* chosen on surface i) contained in a system of conductors j = 1, 2, ..., `*n`*:

ϕ ϕ i = ∑ ∑ j = 1 n 1 4 π π ϵ ϵ 0 ∫ ∫ S j σ σ j d a j R j i (i=1, 2..., n) , {\\displaystyle \\phi _{i}=\\sum _{j=1}^{n}{\\frac {1}{4\\pi \\epsilon _{0}}}\\int _{S_{j}}{\\frac {\\sigma _{j}da_{j}}{R_{ji}}}{\\mbox{ (i=1, 2..., n)}},}

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. σj is not, in general, uniformly distributed across the surface. Let us introduce the factor `*f`*j that describes how the actual `F33f`_`[charge density`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Charge_density]`_`f differs from the average and itself on a position on the surface of the j-th conductor:

σ σ j ⟨ ⟨ σ σ j ⟩ ⟩ = f j , {\\displaystyle {\\frac {\\sigma _{j}}{\\langle \\sigma _{j}\\rangle }}=f_{j},}

or

σ σ j = ⟨ ⟨ σ σ j ⟩ ⟩ f j = Q j S j f j . {\\displaystyle \\sigma _{j}=\\langle \\sigma _{j}\\rangle f_{j}={\\frac {Q_{j}}{S_{j}}}f_{j}.}

Then,

ϕ ϕ i = ∑ ∑ j = 1 n Q j 4 π π ϵ ϵ 0 S j ∫ ∫ S j f j d a j R j i . {\\displaystyle \\phi _{i}=\\sum _{j=1}^{n}{\\frac {Q_{j}}{4\\pi \\epsilon _{0}S_{j}}}\\int _{S_{j}}{\\frac {f_{j}da_{j}}{R_{ji}}}.}

It can be shown that ∫ ∫ S j f j d a j R j i {\\displaystyle \\int _{S_{j}}{\\frac {f_{j}da_{j}}{R_{ji}}}} is independent of the distribution σ σ j {\\displaystyle \\sigma _{j}} . Hence, with

p i j = 1 4 π π ϵ ϵ 0 S j ∫ ∫ S j f j d a j R j i , {\\displaystyle p_{ij}={\\frac {1}{4\\pi \\epsilon _{0}S_{j}}}\\int _{S_{j}}{\\frac {f_{j}da_{j}}{R_{ji}}},}

we have

ϕ ϕ i = ∑ ∑ j = 1 n p i j Q j (i = 1, 2, ..., n) . {\\displaystyle \\phi _{i}=\\sum _{j=1}^{n}p_{ij}Q_{j}{\\mbox{ (i = 1, 2, ..., n)}}.}

>>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 `F33f`_`[system of linear equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=System_of_linear_equations]`_`f is

ϕ ϕ 1 = p 11 Q 1 + p 12 Q 2 ϕ ϕ 2 = p 21 Q 1 + p 22 Q 2 . {\\displaystyle {\\begin{matrix}\\phi _{1}=p_{11}Q_{1}+p_{12}Q_{2}\\\\\\phi _{2}=p_{21}Q_{1}+p_{22}Q_{2}\\end{matrix}}.}

On a `F33f`_`[capacitor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Capacitor]`_`f, the charge on the two conductors is equal and opposite: `*Q`* = `*Q`*1 = -`*Q`*2. Therefore,

ϕ ϕ 1 = ( p 11 − − p 12 ) Q ϕ ϕ 2 = ( p 21 − − p 22 ) Q , {\\displaystyle {\\begin{matrix}\\phi _{1}=(p_{11}-p_{12})Q\\\\\\phi _{2}=(p_{21}-p_{22})Q\\end{matrix}},}

and

Δ Δ ϕ ϕ = ϕ ϕ 1 − − ϕ ϕ 2 = ( p 11 + p 22 − − p 12 − − p 21 ) Q . {\\displaystyle \\Delta \\phi =\\phi _{1}-\\phi _{2}=(p_{11}+p_{22}-p_{12}-p_{21})Q.}

Hence,

C = 1 p 11 + p 22 − − 2 p 12 . {\\displaystyle C={\\frac {1}{p_{11}+p_{22}-2p_{12}}}.}

>>Related coefficients

Note that the array of linear equations

ϕ ϕ i = ∑ ∑ j = 1 n p i j Q j (i = 1,2,...n) {\\displaystyle \\phi _{i}=\\sum _{j=1}^{n}p_{ij}Q_{j}{\\mbox{ (i = 1,2,...n)}}}

can be inverted to

Q i = ∑ ∑ j = 1 n c i j ϕ ϕ j (i = 1,2,...n) {\\displaystyle Q_{i}=\\sum _{j=1}^{n}c_{ij}\\phi _{j}{\\mbox{ (i = 1,2,...n)}}}

where the `*c`*ij with i = j are called the coefficients of capacity and the `*c`*ij with i ≠ j are called the coefficients of electrostatic induction.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

For a system of two spherical conductors held at the same potential,`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]

Q a = ( c 11 + c 12 ) V , Q b = ( c 12 + c 22 ) V {\\displaystyle Q_{a}=(c_{11}+c_{12})V,\\qquad Q_{b}=(c_{12}+c_{22})V}

Q = Q a + Q b = ( c 11 + 2 c 12 + c b b ) V {\\displaystyle Q=Q_{a}+Q_{b}=(c_{11}+2c_{12}+c_{bb})V}

If the two conductors carry equal and opposite charges,

ϕ ϕ 1 = Q ( c 12 + c 22 ) ( c 11 c 22 − − c 12 2 ) , ϕ ϕ 2 = − − Q ( c 12 + c 11 ) ( c 11 c 22 − − c 12 2 ) {\\displaystyle \\phi _{1}={\\frac {Q(c_{12}+c_{22})}{(c_{11}c_{22}-c_{12}^{2})}},\\qquad \\quad \\phi _{2}={\\frac {-Q(c_{12}+c_{11})}{(c_{11}c_{22}-c_{12}^{2})}}}

C = Q ϕ ϕ 1 − − ϕ ϕ 2 = c 11 c 22 − − c 12 2 c 11 + c 22 + 2 c 12 {\\displaystyle \\quad C={\\frac {Q}{\\phi _{1}-\\phi _{2}}}={\\frac {c_{11}c_{22}-c_{12}^{2}}{c_{11}+c_{22}+2c_{12}}}}

The system of conductors can be shown to have similar symmetry `*c`*ij = `*c`*ji.

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Course of Theoretical Physics, Vol. 8), 2nd ed. (Butterworth-Heinemann, Oxford, 1984) p. 4.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citereflekner2011`aLekner, John (2011-02-01). "Capacitance coefficients of two spheres". `*Journal of Electrostatics`*. `!69`! (1): 11–14. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.elstat.2010.10.002.

• `F33f`_`[James Clerk Maxwell`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=James_Clerk_Maxwell]`_`f (1873) `F33f`_`[A Treatise on Electricity and Magnetism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=A_Treatise_on_Electricity_and_Magnetism]`_`f, § 86, page 89.

`c`F0af`_`[↑ Back to top`#top]`_`f`a