Magneostatics - hollow sphere of spontaneous magnetization

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Homework Statement



a) A hollow magnetic sphere of internal radius a and external radius b, has a uniform spontaneous magnetization M per unit volume. SHow that the field strength in the internal cavity (r<a) is zero, and that the external field strength (r>b) is the same as that of a dipole moment [itex] m = 4 \pi M (b^3-a^3)/3 [/itex] , the total moment of the hollow sphere.

b) Show also that the square of the field strength outside the sphere at a point measured from the centre of the sphere and with the respect to the direction of magnetization, is
[tex] H^2 = (3\cos^2\theta +1)({\frac{M(b^3-a^3)}{3r^3}})^2. [/tex]



Homework Equations


[tex] J_b = curl M [/tex]
[tex]K_b = M x n [/tex]
[tex]H=b/\mu_0 - M
[/tex]


The Attempt at a Solution



I am not sure at all how to approach this problem. I am not entirly sure how to use the spontaneous magnetization to model the problem. Any hint of how to look at this problem or any material that would help me learn about the principles involved would b greatly apreciated. Thank yuo for your time.
 

Answers and Replies

  • #2
181
1

Homework Statement



a) A hollow magnetic sphere of internal radius a and external radius b, has a uniform spontaneous magnetization M per unit volume. SHow that the field strength in the internal cavity (r<a) is zero, and that the external field strength (r>b) is the same as that of a dipole moment [itex] m = 4 \pi M (b^3-a^3)/3 [/itex] , the total moment of the hollow sphere.

b) Show also that the square of the field strength outside the sphere at a point measured from the centre of the sphere and with the respect to the direction of magnetization, is
[tex] H^2 = (3\cos^2\theta +1)({\frac{M(b^3-a^3)}{3r^3}})^2. [/tex]


For part a), use principle of superposition. Consider superimposing two sphere of radius b and a of the same magnetization density but in opposite direction.

For part b), shouldn't be too hard once you get part a) since the field of a dipole is well known and can be found in your textbook somewhere.
 

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