- #1

Knut

- 2

- 3

- Homework Statement
- Approximately calculate the voltage U(t) inside a conductive ring.

- Relevant Equations
- none were given

As I`` m learning for an upcoming exam I found an electrodynamics problem I struggle with.

In the first task I need to calculate the magnetic dipole moment of a uniformly charged,thin disk with the Radius R and a total charge Q which rotates with a angular speed omega round its symmetry axis. Which i did in zylinder coordinates:

$$\vec m = \frac{1}{2} \int \vec r \times \vec j(\vec r) d^3 r= \frac{Q}{2 \pi} \int \vec r \times (\vec \omega \times \vec r) d^3 r = \frac{Q \omega R^2}{4} \vec e_z$$

so far so good.

The next task is to take this exact disk and rotate it slowly by the angle theta(t) around the x-axis. There is a conducting ring with the radius r (r<<L) in the xy-plane on the y-axis. It's distance to the disk is L and L>>R. The second task is to approximately calculate the voltage U(t) in the ring.

I thought of calculating the B-field in the z-direction first without an angle. $$\vec B (\vec r) = \frac{\mu_0}{4 \pi} \frac{3 \vec r (\vec m \cdot \vec r ) - \vec m |r|^2}{|r|^5}=\frac{-\mu_0 Q \omega R^2}{16 \pi L^3} \vec e_z$$

The problem begins here as I don't know if I can just add a cos(theta(t)) to the equation to get the right magnetic field that acts on the conductive ring. And then go on with calculating the magnetic flux with $$\Phi = \int_A \vec B \cdot d\vec A$$ and the voltage afterwards with $$U=\frac{d\Phi}{dt}$$ or if I got something wrong by thinking it's that easy. I would be very pleased if someone could help me in that regard.