ρ

J

, are

\nabla \cdot E \nabla \times E = ρ / ϵ_{0}, = - \partial_{t} B, \nabla \cdot B \nabla \times B = 0, = μ_{0} J + μ_{0} ϵ_{0} \partial_{t} E .

(1)

The source-free specialization sets

ρ = 0

and

J = 0

; it cannot describe the charged objects at their locations. Applying the divergence theorem to the first equation gives Gauss's law:

\oint_{\partial Ω} E \cdot d S = \int_{Ω} \nabla \cdot E d^{3} x = \frac{Q _{enc}}{ϵ _{0}} .

(2)

For an isolated point charge

q_{1}

, spherical symmetry makes the electric field radial and constant in magnitude on a concentric sphere. Therefore

4 π r^{2} E_{r} = q_{1} / ϵ_{0}

. Multiplying by a second point charge gives Coulomb's law,

E = \frac{q _{1}}{4 π ϵ _{0} r ^{2}} \hat{r}, F_{2} = \frac{q _{1} q _{2}}{4 π ϵ _{0} r ^{2}} \hat{r} .

(3)

The force points away from the first charge for like signs and toward it for opposite signs.

For the spherical capacitor, a Gaussian sphere between the shells encloses

q

, so

E_{r} = q / (4 π ϵ_{0} r^{2})

for

a < r < b

. Integrating the electric field inward from the outer shell yields

V (r) - V (b) = - \int_{b}^{r} \frac{q}{4 π ϵ _{0} s ^{2}} d s = \frac{q}{4 π ϵ _{0}} (\frac{1}{r} - \frac{1}{b}) .

(4)

An arbitrary constant

V (b)

remains unless a reference is specified. The capacitance is determined by the potential difference:

C = \frac{q}{V ( a ) - V ( b )} = 4 π ϵ_{0} \frac{ab}{b - a} .

(5)

For the subsequent charging operation, take the initial outer shell to be earthed, or interpret the stated

V

as the inner-to-outer voltage with the outer reference zero. Isolating the inner shell fixes its electric charge at

q = C V

. Removing the outer shell leaves an isolated conducting sphere. Its electric field and electric potential, now with zero at infinity, are

E (r) = {0, q \hat{r} / (4 π ϵ_{0} r^{2}), r < a, r > a, V (r) = \frac{q}{4 π ϵ _{0} r} (r \geq a) .

(6)

Consequently

V_{surface} = \frac{bV}{b - a}, V_{old outer position} = \frac{aV}{b - a} .

(7)

These changes of electric potential are compatible with constant electric charge. Without the initial reference convention the absolute stated

V

does not determine

q = C V

: for example, if the outer shell were initially isolated and neutral, the absolute inner electric potential would instead be

q / (4 π ϵ_{0} a)

Finally take the connected spheres to be sufficiently far apart that the long wire's capacitance and their mutual influence can be neglected. Their common electric potential requires

\frac{q _{a}}{a} = \frac{q _{2 a}}{2 a}, q_{a} + q_{2 a} = q,

(8)

q_{a} = q /3

and

q_{2 a} = 2 q /3

. The magnitudes of the surface electric fields are

E_{a} = \frac{q}{12 π ϵ _{0} a ^{2}}, E_{2 a} = \frac{q}{24 π ϵ _{0} a ^{2}}, E_{a} : E_{2 a} = 2 : 1.

(9)

This is the usual widely separated-sphere approximation. A wire enforces an equipotential, but alone does not make this ratio exact for arbitrary arrangements. As a concrete counterexample, if the second shell enclosed the first concentrically and they were connected, all net electric charge would reside on the exterior and the inner sphere's surface electric field would be zero.