Conductors as Boundaries

Section 32.9 Conductors as Boundaries

We have stated above that electric field inside a conductor is zero in the case of electrostatic conditions, and consequently a conductor has the same potential throughout its body, including its surface. Hence, the surface of a conductor is a constant potential surface. Therefore, a conductor having a particular amount of excess charge will be at a definite potential with respect to zero at infinity.

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Equipotential surfaces naturally occur in space around charges and you could cover the surface with a conductor. Covering an equipotential surface with a metal will separate the inside space from the outside space. If you do that the outside world will not notice any difference.

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For instance, an isolated charge has equipotential surfaces in the shape of spherical surfaces centered at the charge. If you cover a particular spherical equipotential surface around the point charge by a spherical conductor with enough charge to match the value of the potential for that equipotential surface, the electric potential and electric field outside the conductor will not change, although the electrical potential and electric field inside the conductor will be different than what had been in the case of the point charge.

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Another more complicated example comes from a spherically-shaped zero potential surface when you have two unequal charges as shown in Figure 32.40. This situation shows that the potential from an electric charge and a spherical conductor at zero potential is same as the potential of two unequal charges and no spherical conductor as long as we look at the field points that are outside the sphere.

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Figure 32.40. The spherical equipotential of a two charge system can be covered by a conductor leaving the potential at all points outside the sphere same as before. The two-charge system in (a) has same potential outside the sphere as a system consisting of one charge and an uncharged sphere have the same potential everywhere outside the sphere. We say that $q_2$ is image of charge $q_1$ in a “spherical mirror”.
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The importance of conductors as equipotential bodies relies in the fact that, once you have set a conductor at some potential, all points of its surface act as a boundary condition of constant value for potential function outside the conductor. This is highly useful due to an important mathematical theorem obeyed by electric potential:

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Uniqueness of potential: Two electric potential functions that satisfy the same boundary conditions must be identical.

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This says that once we have picked a reference for electric potential, only one function of space can satisfy all boundary conditions. Therefore, if you can guess a function that satisfies the potentials at all boundaries in a particular situation, then that guess will be equivalent to the solution of the problem done by other methods. The uniqueness theorem gives a license to guessing - if two problems have the same potential values at all boundary points of a given region, then the two problems give same potentials at all points of the region.

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Example 32.41. Potential Between Two Metallic Spherical Shells.

A spherical metallic shell of outer radius $R_1$ is surrounded by another metallic spherical shell of radius $R_2\text{.}$ The two shells are maintained at different potentials, $V_1$ and $V_2$ respectively? What is the potential at a point P between the two spherical shells?

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Solution.

There are three conditions for potential function $V(r)$ to satisfy here, two at the boundaries and one at the reference point at infinity.

\begin{align*} \amp V(r) = V_1\ \ \ \textrm{when}\ \ r=R_1\\ \amp V(r) = V_2\ \ \ \textrm{when}\ \ r=R_2\\ \amp V(r) = 0\ \ \ \textrm{when}\ \ r=\infty \end{align*}

The following function would satisfy these conditions and hence gives potential at any point between the two shells.

\begin{equation*} V(r) = \left( \frac{R_1R_2}{R_2-R_1}\right)\left[ V_1\left( \frac{1}{R_1}-\frac{1}{r} \right) + V_2\left( \frac{1}{r}-\frac{1}{R_2} \right) \right] \end{equation*}

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