Question about concentric conductors

[Jason Williams]

Homework Statement

Three insulated concentric spherical conductors, whose radii in ascending order of magnitude are $a, b, c$, have charged $e_1, e_2, e_3$ respectively, find their potentials and show that if the inner most sphere is grounded, the potential of the outermost is diminished by:

$\frac{a}{c} ( \frac{k e_1}{a} + \frac{k e_2}{b} + \frac{k e_3}{c} )$.

Homework Equations

$k = \frac{1}{4 \pi \epsilon_0}$

The Attempt at a Solution

[/B]
The potentials in each region are $V_I = \frac{k e_1}{r} + C$, $V_{II} = \frac{k (e_1 + e_2)}{r} + D$, and $V_{III} = \frac{k (e_1 + e_2 + e_3)}{r} + E$, where $C, D$ and $E$ are integration constants. Setting the potential at the innermost sphere to $0$, we solve for $C$, giving us $V_{III} = \frac{k e_1}{r} - \frac{k e_1}{a}$. Ensuring continuity @ $b$ and $c$, I finally get $V_{III} = \frac{k (e_1 + e_2 + e_3)}{r} - (\frac{e_3}{c} + \frac{e_2}{b} + \frac{e_1}{a})$. I just don't see how we get the factor of $\frac{a}{c}$ on the outside. Help is greatly appreciated.

 

[BvU]

Hello Jason,

I read the exercise as consisting of two parts: a) find the potentials, and b)ground the inner sphere and ... etc.

 

[Jason Williams]

Hello Jason,

I read the exercise as consisting of two parts: a) find the potentials, and b)ground the inner sphere and ... etc.

Oops, sorry you're totally right. I left the potentials in that form because you can't solve for $C$, $D$, and $E$ without knowing the potentials on each surface. Is this the wrong way of approaching the problem?

 

[BvU]

you should be able to solve part a using potental zero at infinity