Magnetic Materials — Question 2

Question 2

A long linear core separates free-current and bound-current contributions

A very long solenoid has n turns per unit length, current I, and a homogeneous linear core of susceptibility chi. Neglect end effects and hysteresis.

1. Find H, M, B and the equivalent bound surface-current magnitude on the cylindrical core.

2. Find the reversible magnetic energy density and its ratio to the empty-solenoid value at the same current.

Question 2 - Solution

A long linear core separates free-current and bound-current contributions

(a) Method and calculation

$$H=nI,\quad M=\chi nI,\quad\boxed{B=\mu_0(1+\chi)nI}.$$$$\mathbf K_b=\mathbf M\times\hat n\Rightarrow\boxed{|K_b|=|\chi nI|}.$$

(b) Interpretation and further calculation

$$u=\int_0^B H(B^{\prime})\,dB^{\prime}=\frac{B^2}{2\mu_0(1+\chi)}=\frac12BH.$$$$\boxed{u/u_0=1+\chi}\quad\text{at fixed }I.$$

Physical check

H is set by free current in this long-solenoid geometry; magnetization supplies an additional contribution to B.