# Concentric conducting spherical shells

1. Jan 8, 2014

### unscientific

1. The problem statement, all variables and given/known data

Consider three concentric conducting shells, with potentials $$0, ø_0, 0$$ and radius $$a, b ,c$$ where $$a < b < c$$.

(a)State conditions for Laplace to work and boundary conditions for E

(b)Show ø is of the form:

(c) Find ø and E everywhere.

(d) Find the charge density and electrostatic energy.

2. Relevant equations

3. The attempt at a solution

Part (a)

Condition for laplace: Potential at surface of shells must be constant
Boundary Conditions for E: As $$r → \infty |\vec {E}| → 0$$

Part (b)

Consider $$r → \infty , V → const.$$

$$\sum_{l=0}^{\infty} (A_l r^l)P_{l cosθ} → const.$$

This implies that $$l = 0$$ is the only solution.

Thus, general solution for ø:
$$ø = A + \frac {B}{r}$$

Part (c)
1. ø must be continuous at r= a, so $$ø_{in} = ø_{out}$$ at r=a.
2. Potential at r = a is 0.
We obtain two simultaneous equations:
$$A = \frac {B}{a}$$
$$0 = A + \frac {B}{a}$$
So this implies $$A = B = 0$$ for the first shell.

Middle shell
At r = b, $$ø_0 = A + \frac {B}{a}$$
Continuity: $$ø_{in} = ø_{out} so [tex]Ab = B$$
Solving, we get $$ø_{in} = \frac {ø_0}{2}, ø_{out} = \frac{bø_0}{2r}$$

Outermost Shell
Does nothing, as shown above.

Using superposition principle, so far we have:
$$ø = \frac {ø_0}{2} (0 < r <b)$$
$$ø = \frac {ø_0b}{2r} (r > b)$$

Using $$E = - \nabla ø$$

$$E = 0 (0 < r < b)$$
$$E = \frac {bø_0}{2r^2} (r > b)$$

Graphs

Part(d)

At $$r = b, \vec{E} = -\nabla V$$
$$(\frac {\partial {V_{out}}}{\partial {r}} - \frac{\partial {V_{in}}}{\partial {r}}) = -\frac {σ_b}{ε_0}$$
$$(\frac {-bø_0}{2b^2}) = -\frac {σ}{ε_0}$$
$$σ_b = \frac {ø_0ε_0}{2b}$$

For r = a,
$$(\frac {\partial {V_{a,out}+V_{b,in}}}{\partial r} - \frac {\partial {V_{a,out}+V_{b,in}}}{\partial r}) = 0$$
So $$σ_a = 0$$

$$Energy = \frac{1}{2} ε_0 \int_{a}^{c} E^2 dV = \frac {1}{2} ε_0 \int_b^c \frac {bø_0}{2r^2} dr = \frac {1}{4}ε_0bø_0(\frac {c-b}{bc})$$

Last edited: Jan 8, 2014
2. Jan 9, 2014

### unscientific

bumpp

3. Jan 11, 2014

bumppp