Ferromagnetism — Question 3

Question 3

A rectangular magnetization loop converts its area into heat

An ideal hysteresis cycle switches abruptly between M=-Ms and M=+Ms at internal fields +Hc and -Hc, respectively. Outside the switches M is constant. Use the loss per unit volume $w=\oint H\,dB$, with $B=\mu_0(H+M)$.

1. Derive w from the two switching events.

2. Find average loss power for sample volume V cycled at frequency f, and evaluate it for $M_s=8.0\times10^5$ A/m, $H_c=100$ A/m, $V=2.0\times10^{-5}$ cubic meters, and f=50 Hz. Use $\mu_0=4\pi\times10^{-7}$ H/m.

Question 3 - Solution

A rectangular magnetization loop converts its area into heat

(a) Method and calculation

$$w=\mu_0\oint H\,dH+\mu_0\oint H\,dM=\mu_0\oint H\,dM.$$$$w=\mu_0[(H_c)(2M_s)+(-H_c)(-2M_s)]=\boxed{4\mu_0H_cM_s}.$$

(b) Interpretation and further calculation

$$\boxed{P=fVw=4\mu_0fVH_cM_s}.$$$$w=402.124\,\mathrm{J/m^3},\qquad\boxed{P=0.4021\,\mathrm W}.$$

Physical check

This quasistatic loop-area loss excludes additional eddy-current and dynamic losses.