Putting an Electron on a Conducting Sphere

Question

What all might go on inside a conducting sphere if I (try to) dump an electron upon it?

Answer 1

If you were holding some charge there with some force and always had then an equal charge would distribute throughout the surface of the conductor so that an equal but opposite charge could be right where you are holding your charge. So it is just like the charge was always distributed on the surface.

If however you inserted some charge somewhere really fast and then let go, then things would start to change and so it isn't electrostatics anymore. Or if you held that charge and all the other charges too and then let go then it would not be electrostatics anymore but then at least the fields would start out electrostatic since you were holding everything static.

You can often think of a good conductor as a material with $\vec J =\sigma \vec E$ and find out the state it approaches after an infinite amount of time. And if you let $\sigma$ get large it approaches it quickly. In that case your object gets super close to that in a very short time.

So take $\vec J=\sigma \vec E$ then you can take the divergence of both sides and get $$\sigma\frac{\rho}{\epsilon_0}=\sigma\vec \nabla \cdot \vec E = \vec \nabla \cdot \vec J$$

Where we used the Maxwell equation $\dfrac{\rho}{\epsilon_0}=\vec \nabla \cdot \vec E$ and we can also take the divergence of $$\vec \nabla \times \vec B=\mu_0\vec J+\mu_0\epsilon_0\frac{\partial \vec E}{\partial t}$$ to get the continuity equation

$$\vec \nabla \cdot \vec J=-\epsilon_0\vec \nabla \cdot\frac{\partial \vec E}{\partial t}=-\frac{\partial \rho}{\partial t}.$$

This means we have $$\frac{\partial \rho}{\partial t}=-\vec \nabla \cdot \vec J=-\frac{\sigma}{\epsilon_0}\rho.$$

This means that if you insert some net charge into a conductor the amount of net charge imbalance decays exponentially fast. And if $\sigma$ is large that happens super fast. And current flows along the electric field lines. So it flows from positive charge imbalance all the way to either a negative charge imbalance (as they help each other get close to a charge balance $\rho=0$) or to the surface where a charge imbalance develops. And charge can and does also flow from one part of the surface to another if the field line just goes from one to the other without hitting a charge imbalance where the field diverges.

And it can even do that through a region of perfectly balanced charge. It's like if you had people that owed just as much money as they had in cash. They could each borrow a dollar from the person to their left and loan a dollar to the person on their right and they still owe as much as they have in cash except for those people on the two ends. Charge can flow from one point on the surface to another point on the surface in the same way they each push current along a field line and the two ends where the field line hits the surface are left with a charge imbalance.

So the charge inside can decay exponentially fast with a very very fast decay and the charge on the surface cab distribute itself. And it keeps doing that, but the amount of current gets smaller and smaller as the fields inside get smaller and smaller.

Now a conductor might not have $\vec J=\sigma\vec E,$ but the end that material becomes is what a perfect conductor is also aiming for. Basically a small field makes a huge current so huge amounts of charge can distribute away from inside to the outside and from one part of the outside surface to another part of the surface so the charge gets to that final configuration (or really close to it) really fast.