Ferromagnetism — Question 4

Question 4

An elliptical induction loop has a loss fixed by its phase lag

In a steady cycle an internal field and induction are $H=H_0\cos\theta$ and $B=B_0\cos(\theta-\delta)$, with theta increasing from 0 to 2pi and $0<\delta<\pi/2$. Use $w=\oint H\,dB$.

1. Evaluate the cycle loss per unit volume explicitly.

2. Find the ratio of losses for phase lags 30 and 60 degrees at fixed amplitudes, and the limit as delta approaches zero.

Question 4 - Solution

An elliptical induction loop has a loss fixed by its phase lag

(a) Method and calculation

$$w=-H_0B_0\int_0^{2\pi}\cos\theta\sin(\theta-\delta)\,d\theta.$$$$\boxed{w=\pi H_0B_0\sin\delta}.$$

(b) Interpretation and further calculation

$$\boxed{w_{60}/w_{30}=\sin60^\circ/\sin30^\circ=\sqrt3}.$$$$\delta\to0\Rightarrow\boxed{w\to0}\quad\text{as the ellipse collapses to a line}.$$

Physical check

The integral has units A/m times tesla, equivalent to joules per cubic meter; a finite phase lag encloses positive area.