Faraday's Law of Induction — Question 3

Question 3

Rotation creates an alternating emf even when the magnetic field is steady

A coil of N turns and area A rotates in uniform B so its normal makes angle $\theta=\omega t$ with the field. The resistive load is R; ignore inductance and keep angular speed constant.

1. Find emf and its root-mean-square value over one rotation.

2. Find the average load power and identify the phase relation between flux and emf.

Question 3 - Solution

Rotation creates an alternating emf even when the magnetic field is steady

(a) Method and calculation

$$\Lambda=NBA\cos\omega t\Rightarrow\boxed{\mathcal E=NBA\omega\sin\omega t}.$$$$\mathcal E_{\mathrm{rms}}=\sqrt{\frac1T\int_0^T\mathcal E^2dt}=\frac{NBA\omega}{\sqrt2}.$$

(b) Interpretation and further calculation

$$\boxed{\langle P\rangle=\frac{(NBA\omega)^2}{2R}}.$$$$\Lambda\text{ maximal at }\omega t=0;\quad\mathcal E\text{ maximal at }\omega t=\pi/2.$$

Physical check

Maximum flux is not maximum emf; emf depends on the rate of change of flux.