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## 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

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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.