I am confused about the solving for the E field inside of a hollow sphere with a non uniform charge distribution. I understand conceptually that the unbalanced charge distribution would lead to an E-field inside of the sphere, but if I apply Gauss' law, there is no charge enclosed, so it comes out to be E = 0. Does Gauss' law not apply in this situation, or am I using it wrong? And if it does not apply, why?
1 Answer
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Gauss's law will apply. According to the Gauss's law, because there is no net charge inside, the divergence of E will be zero (or if you use the integral form, the net flux of E is zero), but that does not mean E=0.
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$\begingroup$ So how do we differentiate when it applies to flux or when it applies numerically as well, when solving problems? Because if this was a solid sphere, I could apply Gauss' law and get the correct answer. How do I know when it mathematically applies? $\endgroup$ Commented Jan 24, 2019 at 3:23
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$\begingroup$ As a rule of thumb, it's best used with situations that have a high degree of symmetry. If your charge distribution were uniform, or there were some pattern to it, Gauss' Law would be perfect, even the flux version. $\endgroup$ Commented Jan 24, 2019 at 3:45
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$\begingroup$ Agree. if it is non-uniform without any pattern or symmetry, then probably you you will need to numerically solve, say, the Poisson's equation; then take the gradient to get the field... $\endgroup$ Commented Jan 24, 2019 at 3:48