- #1

baseballfan_ny

- 91

- 23

- Homework Statement:
- An insulating sphere of radius R1 has charge density p (rho) uniform, except for a small, hollow region of radius R2 located at a distance a from the center. Find the electric field at a point P inside the hollow region.

- Relevant Equations:
- Gauss' Law, superposition

Here's an image. O and O' are the respective centers, a is the distance between them, r is the distance from the center of the sphere to P, and r' = r - a, the distance from O' to P.

The approach (which I don't understnad) given is to use Gauss' Law and superposition, so that we calculate the electric field at P if there was no hollow sphere and get,

##\iint_{Closed Surface} \vec E \cdot d\vec A = \frac {Q_{enc}} {\epsilon_0}##

## E 4\pi*r^2 = \frac {\rho * \frac {4} {3} \pi*r^3} {\epsilon_0}##

## E = \frac {\rho r} {3\epsilon_0}##

Then the hollow sphere is treated as a region of "negative charge density" ##\rho##, and we calculate E at P from O' (at a distance r'), so we get ##E = -\frac {\rho r'} {3\epsilon_0}##

And we add the two together to find ##E_P = \frac {\rho r - r'} {3\epsilon_0} = \frac {\rho a} {3\epsilon_0}##.

This approach really confuses me and I am not sure why it works. I get that we have to subtract some charge out given that there's an empty sphere, but I do not understand why we do so with a sphere of radius r' around the center of the hollow sphere. For some clarification, to me it looks like that when I draw a Gaussian surface through Point P, there's not enough symmetry to apply Gauss' Law, as shown in the picture below:

Then I would assume we need to subtract the region I've shaded in above (the intersection between the Gaussian surface and the hollow sphere), as that's the enclosed region where there's "missing charge" and is causing the "disruption in symmetry." But the approach is to instead subtract a region of radius r' from the center of the hollow sphere, which is the shaded region illustrated in the image below:

I really am having trouble understanding why we have to subtract this region rather than the previous one. Are the two geometrically equivalent? Or am I totally missing something about satisfying the symmetry for the Divergence Theorem?

Thank you so much in advance!