Angular velocity of a spherical capacitor

[quasar_4]

Homework Statement

A spherical capacitor made out of two concentric conducting shells of radii a
and b is placed in homogeneous magnetic field H0, and is allowed to rotate freely (without friction) around the axis parallel to this magnetic field. Initially, there are charges Q
Q− /Q+ , on the surfaces of capacitor. What will be the steady-state angular velocity of this capacitor, if its moment of inertia is I? Assume that discharging of capacitor occurs without external forces.

Homework Equations

L=I*omega, where omega = angular velocity
Maxwell's equations

The Attempt at a Solution

I am sooo confused by this problem. The shells are conducting, and the way the problem is worded seems to indicate that the capacitor is discharging, so obviously charges are free to move. First of all, I can see no reason why this sphere would begin spinning... but fine, let's just imagine that I somehow got it started. The charges moving around would resemble a current in the phi direction. And there would be some sort of induced magnetic field. Maybe I could think of this as some sort of inductance problem... if I could find the current in one shell I could try to compute the magnetic flux through the other shell. But I'm perplexed as to what's physically happening and where to even begin. If I had some starting point maybe I could find the force on the capacitor, a sort of F = q v x B thing, and from that, find v, and from that, omega and hence the angular momentum... but I don't know where to begin! Help!

 

[mark726]

Thank you for your question. It is understandable that this problem may seem confusing at first, but with some careful analysis, we can find a solution. Let's break down the problem step by step.

First, let's consider the initial state of the capacitor. Since it is placed in a homogeneous magnetic field, there will be a force acting on the charges on the surface of the capacitor, causing them to move. This movement will create a current in the phi direction, as you correctly mentioned. This current will in turn create a magnetic field, which will interact with the external magnetic field. This interaction will result in a torque on the capacitor, causing it to start rotating.

Next, let's consider the discharging of the capacitor. As the charges move around, they will eventually reach a steady state, where the rate of charge transfer (discharging) is equal to the rate of charge accumulation. In this steady state, the current in the phi direction will also reach a steady state, and the torque on the capacitor will be constant. This means that the angular velocity, omega, will also reach a steady state.

Now, let's apply Maxwell's equations to this situation. We can use Faraday's law, which states that the induced electric field is equal to the negative of the time rate of change of the magnetic flux. In this case, the magnetic flux through the capacitor will be changing due to the rotation of the capacitor. Using this equation, we can find the induced electric field, which will in turn give us the current in the phi direction.

Finally, we can use the relationship L=I*omega to find the steady-state angular velocity, where L is the angular momentum and I is the moment of inertia. We know the moment of inertia from the given information, and we can calculate the angular momentum from the torque and the time it takes for the capacitor to reach the steady state.

I hope this helps you understand the problem better and gives you a starting point for finding a solution. Don't hesitate to ask for clarification if needed. Good luck!