# Gauss's law unclear here

Walter Lewin in one of his lectures (8.02 lect 5 https://youtu.be/JhV-GOS4y8g?t=25m46s) draws an example of a solid conducting heart.

He then asks whether there would be any charge on the inner surface of the heart or all the charge would be located on the outer surface. He uses a gaussian surface (red) to prove that there cannot be any charge on the inner surface, enclosed within the gaussian surface, since the net flux and therefore electric field is 0.

My question is, how did he come to the conclusion that there can't be any charge on the inner surface? Isn't there a possibility that the charges would align themselves to cancel the fields within the cavity, but also in the area between the two surfaces? Like this (sorry for the messy image):

So as you probably already know, one of the properties of a conductor is that there cannot be any non-zero electric field inside the conducting material. Let us consider there to be a Gaussian surface inside of the heart-shape conductor (i.e. $\oint \vec E \cdot d\vec a = Q_e /\epsilon_0$). Since there cannot be any electric field, the left hand side is $0$. So therefore the charge enclosed, $Q_e$, must be $0$ as well for the right hand side to be $0$. This means one of these two things:

1. There are no charge inside of the cavity, thus $Q_e = 0$.
2. There are some charges inside of the cavity, but the conductor rearranges its free charges to cancel out the charges inside of the cavity, i.e. $Q_e = q_c + q_f = 0$ where $q_c$ is the charge inside of the cavity and $q_f$ is the charge that the conductor rearranges to cancel out $q_c$. Now, the way the conductor arrange $q_f$ is to distribute it uniformly around the inner surface.

In the lecture, since Lewin did not draw any charge inside of the cavity, it means that scenario 1. is what he meant. Since there are no charge inside of the cavity, there cannot be any charge on the inner surface because if there were charges on the inner surface the RHS of Gauss's law would then be non-zero implying the electric field inside of the conductor is non-zero as well.

Note: Please let me know if I misunderstood what you are trying to ask.

• Hey, thanks for the concise answer. I was asking more on the conceptual level why there must be charge enclosed in this scenario(without using Gauss law equation). Why can't there be + charge in the inner surface arranged just in the right way to cancel out the field in the inner cavity, and the + charge in the outer surface arranged just in the right way to cancel out the presence of the inner + charges' electric field between the inner and outer surfaces? This would mean there are charges inside the gaussian surface, but the E is zero – Ivan Simunovic Jan 15 '18 at 6:37

There would be a non-zero electric field inside the heart if there were charges on the inner boundary. As you know, there can be no electric field inside a conductor.

• But they could again rearrange themselves to cancel the field inside the heart, just like charges within a hollow sphere surface do, no? – Ivan Simunovic Jan 14 '18 at 19:58
• Then there would be an electric field between the inner and outer surface. – Thierry Kauffmann Jan 14 '18 at 20:30
• But that can be canceled out too by the charges on the surfaces – Ivan Simunovic Jan 14 '18 at 20:51