Suppose the field inside a large piece of magnetic material is B0, so that H0 = (1/μ0)B0 − M, where M is a “frozen-in” magnetization. (a) Now a small spherical cavity is hollowed out of the material (Fig. 6.21). Find the field at the center of the cavity, in terms of B0 and M. Also find H at the

Question

Suppose the field inside a large piece of magnetic material is [latex]B _{0}[/latex] , so that [latex]H _{0}=\left(1 / \mu_{0} ight) B _{0}- M[/latex] , where M is a “frozen-in” magnetization.

(a) Now a small spherical cavity is hollowed out of the material (Fig. 6.21). Find the field at the center of the cavity, in terms of [latex]B _{0}[/latex] and M. Also find H at the center of the cavity, in terms of [latex]H _{0}[/latex] and M.

(b) Do the same for a long needle-shaped cavity running parallel to M.

(c) Do the same for a thin wafer-shaped cavity perpendicular to M.

Assume the cavities are small enough so M, [latex]B _{0}, ext { and } H _{0} [/latex] are essentially constant. Compare Prob. 4.16. [Hint: Carving out a cavity is the same as superimposing an object of the same shape but opposite magnetization.]

Accepted Answer

(a) The field of a magnetized sphere is [latex]\frac{2}{3} \mu_{0} M [/latex] (Eq. 6.16), so [latex]B = B _{0}-\frac{2}{3} \mu_{0} M[/latex] , with the sphere removed.

In the cavity, [latex]H =\frac{1}{\mu_{0}} B , ext { so } H =\frac{1}{\mu_{0}}\left( B _{0}-\frac{2}{3} \mu_{0} M ight)= H _{0}+ M -\frac{2}{3} M \Rightarrow H = H _{0}+\frac{1}{3} M[/latex] .

(b) The field inside a long solenoid is [latex]\mu_{0} K[/latex]. Here K = M, so the field of the bound current on the inside surface of the cavity is [latex]\mu_{0} K[/latex], pointing down. Therefore

[latex]B = B _{0}-\mu_{0} M[/latex] .

[latex]H =\frac{1}{\mu_{0}}\left( B _{0}-\mu_{0} M ight)=\frac{1}{\mu_{0}} B _{0}- M \Rightarrow H = H _{0}[/latex] .

(c) This time the bound currents are small, and far away from the center, so [latex]B = B _{0}[/latex] , while [latex]H =\frac{1}{\mu_{0}} B _{0}= H _{0}+ M \Rightarrow H = H _{0}+ M[/latex] .

[Comment: In the wafer, B is the field in the medium; in the needle, H is the H in the medium; in the sphere (intermediate case) both B and H are modified.]

Question 6.13: Suppose the field inside a large piece of magnetic material ...

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