Coefficients of potential explained

In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric:

\begin{matrix} \phi _{1} = p _{11} Q _{1} + … + p _{1n} Q _{n} \\ \phi _{2} = p _{21} Q _{1} + … + p _{2n} Q _{n} \\ \vdots \\ \phi _{n} = p _{n1} Q _{1} + … + p _{nn} Q _{n \end{matrix}.}

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 _{21} " is the due to charge 1 on conductor 2.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. , by symmetry, and
2. 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 ^{n} _{j =1} (1)/(4 \pi\epsilon _{0}) \int _{S _{j}} (\sigma _{j} da _{j})/(R _{j}) .

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 ^{n} _{j =1} (1)/(4 \pi\epsilon _{0}) \int _{S _{j}} (\sigma _{j} da _{j})/(R _{ji}) (i=1,2...,n) ,

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:

(\sigma _{j})/(\langle\sigma _{j\rangle}) = f _{j,} or\sigma _{j} = \langle\sigma _{j\rangle} f _{j} = (Q _{j})/(S _{j}) f _{j.} Then,\phi _{i} = \sum ^{n (Q _{j})/(4 \pi\epsilon _{0S} _{j})} _{j =1} \int _{S _{j}} (f _{j} da _{j})/(R _{ji}) . It can be shown that \int _{S _{j}} (f _{j} da _{j})/(R _{ji}) is independent of the distribution \sigma _{j} . Hence, with p _{ij} = (1)/(4 \pi\epsilon _{0} S _{j}) \int _{S _{j}} (f _{j} da _{j})/(R _{ji}) , we have\phi _{i=\sum} ^{n} ^{j =1} p _{ij} Q _{j} (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 system of linear equations is

\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 capacitor, the charge on the two conductors is equal and opposite: . Therefore,

\begin{matrix} \phi _{1} = (p _{11} - p _{12} )Q \\ \phi _{2} = (p _{21} - p _{22} )Q \end{matrix}, and\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}) .

Related coefficients

Note that the array of linear equations

\phi _{i} = \sum ^{n} _{j =1} p _{ij} Q _{j} (i=1,2,...n) can be inverted toQ _{i} = \sum ^{n} _{j =1} c _{ij} \phi _{j} (i=1,2,...n) where the with are called the coefficients of capacity and the with are called the coefficients of electrostatic induction.^[1]

For a system of two spherical conductors held at the same potential,^[2]

Q _{a=(c} _{11} + c _{12} )V , Q _{b=(c} _{12} + c _{22} )V Q = Q _{a+Q} _{b} = (c _{11} +2 c _{12} + c _{bb} )V

If the two conductors carry equal and opposite charges,

\phi _{1= (Q(c _{12} + c _{22} ))/({(c _{11} c _{22} - c ^{2 )} _{12})} }, \qquad \quad \phi_2=\frac C = (Q)/(\phi _{1-\phi} _{2}) = (c _{11} c _{22} - c ^{2} _{12})/(c _{11} + c _{22} +2 c _{12})

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

References

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

Notes and References

1. 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.
2. Lekner. John. 2011-02-01. Capacitance coefficients of two spheres. Journal of Electrostatics. 69. 1. 11–14. 10.1016/j.elstat.2010.10.002.