Magnetic Materials — Question 6

Question 6

Layer orientation changes the effective magnetic permeability

A laminate contains volume fractions f and 1-f of linear media with permeabilities mu1 and mu2. Layers are much thinner than the sample; neglect edges and free interface currents.

1. Derive effective permeability for a uniform field parallel to the layers.

2. Derive it for a field normal to the layers and evaluate both for f=1/2 and mu2=9mu1.

Question 6 - Solution

Layer orientation changes the effective magnetic permeability

(a) Method and calculation

$$H_{1t}=H_{2t}=H\Rightarrow\langle B\rangle=[f\mu_1+(1-f)\mu_2]H.$$$$\boxed{\mu_\parallel=f\mu_1+(1-f)\mu_2=5\mu_1}.$$

(b) Interpretation and further calculation

$$B_{1n}=B_{2n}=B\Rightarrow\langle H\rangle=B[f/\mu_1+(1-f)/\mu_2].$$$$\boxed{\mu_\perp=[f/\mu_1+(1-f)/\mu_2]^{-1}=1.8\mu_1}.$$

Physical check

The arithmetic and harmonic means differ because continuity fixes different field components in the two orientations.