Question

How do I use Gauss' Law to show that the electric field of any point inside (r 13 years ago

Answer 1

The simple answer is because charges are present on the outer surface of the hollow sphere.

The mathematical proof of this is as follows:

Gauss' Law states:
\[\int\limits_{}^{}E*dA=\frac{ q^{enc} }{ \epsilon _{0} }\]In most cases, Gauss' Law is useful because the electric field is constant, and so the integral simplifies immensely:
\[\int\limits_{}^{}E*dA=E \int\limits_{}^{}dA=EA=\frac{ q^{enc} }{ \epsilon _{0} }\]Now the one thing we know is that the total charge enclosed is 0! There are no charges within the hollow sphere. Therefore:
\[EA=\frac{ q^{enc} }{ \epsilon _{0} }=\frac{ 0 }{ \epsilon _{0} }=0\]So now, we have the expression:
\[EA = 0\]And since there is obviously a positive value of area (non-zero), E has to be 0.

Thus, we have proven conceptually and mathematically how Gauss' Law describes the interior of a hollow sphere as having E=0.

Answer 2

@Pompeii00 Thank you! I understand now why it's 0.