Calculating EMF from Rotating Spherical Shell in a Magnetic Field

[ehrenfest]

Homework Statement

A perfectly conducting spherical shell of radius 'a' rotates about the z-axis with angular velocity $\omega$ in a uniform magnetic field $\mathbf{B} = B_0 \hat{\mathbf{z}}$. Calculate the emf developed between the "north pole" and the equator.

Homework Equations

The Attempt at a Solution

I don't understand how emf makes sense in this context. I have only seen emf in currents. The emf is defined as an integral around a LOOP:
$$\int_{loop} \mathbf{f} \cdot d\mathbf{l}$$
I don't see how you would generalize it here.
I already read this https://www.physicsforums.com/showthread.php?t=112819.

 

[Jhero]

Hello there,

Thank you for your question. I can understand your confusion about the concept of emf in this context. Let me try to explain it to you.

Firstly, emf (electromotive force) is a measure of the energy per unit charge that is available from an electric source. In this case, the rotating spherical shell can be considered as an electric source, since it is moving in a magnetic field. The rotation of the shell causes a change in the magnetic flux through the surface, which in turn induces an electric field and hence an emf.

Now, in order to calculate the emf, we need to consider the path along which the emf is being measured. In this case, the path is from the "north pole" to the equator. This path can be thought of as a loop, and hence we can use the same formula for calculating the emf as in a current loop.

So, to calculate the emf, we need to find the line integral of the electric field along this path. The electric field can be calculated using Faraday's law, which states that the emf induced in a closed loop is equal to the rate of change of magnetic flux through the loop.

I hope this explanation helps to clarify the concept of emf in this context. Let me know if you have any further questions.