Induction and Energy Transfers — Question 9

Question 9

A rotating annular conductor generates radial emf and an opposing torque

An annular conducting disk rotates about +z at angular speed omega in field $B\hat z$. Sliding contacts at radii a and b, with b>a, connect a load of resistance R. Ignore contact and disk resistance and self-inductance.

1. Find outer-minus-inner emf and load current.

2. Find the required drive torque from power balance.

Question 9 - Solution

A rotating annular conductor generates radial emf and an opposing torque

(a) Method and calculation

$$\vec v\times\vec B=\omega rB\hat r,\quad\mathcal E=\int_a^b\omega Br\,dr=\frac{B\omega}{2}(b^2-a^2).$$$$\boxed{i=\frac{B\omega(b^2-a^2)}{2R}}.$$

(b) Interpretation and further calculation

$$\tau_{\mathrm{drive}}\omega=\mathcal Ei=i^2R.$$$$\boxed{\tau_{\mathrm{drive}}=\frac{B^2\omega(b^2-a^2)^2}{4R}}.$$

Physical check

The outer contact is positive for the stated rotation and field; the drive must replace the electrical power delivered to the load.