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This question is motivated by Problem 3.4 in Griffiths' book on Electrodynamics.

This problem asks to calculate the average electric field on a sphere, due to charges inside the sphere and outside the sphere. For simplicity, let's just considering a point charge $q$ with field $$\mathbf{E}=\frac{1}{4\pi\epsilon_0}\left(\frac{q}{r^2} \right)\widehat{\mathbf{r}}.$$

If the charge is inside the sphere, then we have $$\mathbf{E}_{\mathrm{avg}}=\frac{1}{4\pi\epsilon_0}\left(\frac{q}{r^2} \right)\widehat{\mathbf{r}},$$ while if the charge is outside then we have $$\mathbf{E}_{\mathrm{avg}}=\mathbf{0}.$$

I understand the mathematical derivation of these results, but am having trouble coming up with a physical interpretation. The first result (outside the sphere) is elegant, but I cannot see why it should be true physically. I am also unable to come up with a physical interpretation of the second result, and a pointer in the right direction would be greatly appreciated.

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Think about the electric field as field lines. Each field line emanates radially from the point charge. Imagine field lines pointing out in every direction, isotropically around the charge.

Now, if you want to calculate the total electric field on a surface, you’re essentially adding every lines that goes through the surface in the “positive” direction, and subtracting every line that goes through in the “negative” direction. So for the sphere, every line going out of the sphere is added, and every line going in is subtracted.

Finally, think about a sphere in the field of a point charge. If the charge is outside the sphere, all of the lines which enter the sphere just keep going on through, and they exit as well. Thus, for every “+”, there’s a “-“, and the total is zero. But when the charge is inside the sphere, the lines originate within, so there is only “+”. Therefore, the net field on the sphere in that case is just the total field emitted from the charge (no matter where the charge is positioned within the sphere).

This trick will help your intuition in other cases as well. Just remember (from Gauss’ Law), field lines begin at positive charges and terminate at negative charges. If there are more positive than negative charges, then the field lines go out to infinity.

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