How Does Potential Work Again?

Electricity and Magnetism Level 3

There is a sphere of radius $10$ centered on the origin of the $xyz$ coordinate system. Within the sphere, there is a uniform volumetric charge density of $1$ .

The region within the sphere is non-conductive, and its electric permittivity is $\epsilon$ . The region outside the sphere is uncharged.

Find the electric potential difference (voltage) $(V)$ between points $(0,0,-5)$ and $(1,2,3)$ . Give your answer as $|\epsilon \,V |$ .

The answer is 1.833.

1 solution

Otto Bretscher
Dec 14, 2018

We need to solve the Poisson equation $\nabla^2 \Phi=\frac{1}{\epsilon}$ , or , in spherical coordinates, $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\Phi}{dr}\right)=\frac{1}{\epsilon}$ , so $\Phi=\frac{r^2}{6\epsilon}$ . For the given points, we find $|\epsilon V| = \frac{11}{6}\approx \boxed{1.833}$ .

How Does Potential Work Again?

The answer is 1.833.

1 solution

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