Induction and Energy Transfers — Question 5

Question 5

A resistively loaded rotating coil exerts a speed-dependent braking torque

A coil rotates at constant angular speed omega in uniform B. It has N turns, area A and total circuit resistance R. Its flux linkage is $NBA\cos\omega t$; neglect inductance.

1. Find instantaneous mechanical power required from the drive.

2. Find the average opposing torque and the work supplied in one rotation.

Question 5 - Solution

A resistively loaded rotating coil exerts a speed-dependent braking torque

(a) Method and calculation

$$\mathcal E=NBA\omega\sin\omega t,\quad i=\mathcal E/R.$$$$\boxed{P_{\mathrm{drive}}=i^2R=(NBA\omega)^2\sin^2\omega t/R}.$$

(b) Interpretation and further calculation

$$\boxed{\langle\tau_{\mathrm{brake}}\rangle=\langle P\rangle/\omega=(NBA)^2\omega/(2R)}.$$$$W_{\mathrm{turn}}=\langle P\rangle\frac{2\pi}{\omega}=\boxed{\pi(NBA)^2\omega/R}.$$

Physical check

The stated torque is its opposing magnitude averaged at constant speed; the instantaneous load varies over the rotation.