Find the magnetic field inside the cylinder

  • #1
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Homework Statement


There's a very long cylinder with radius ##R## and magnetic permeability ##\mu##. The cylinder is placed in uniform magnetic field ##B_{0}## pointed perpendicularly to the axis of cylinder. Find magnetic field for ##r < R##. Assume there's a vacuum outside the cylinder.

Homework Equations


Boundary conditions:
$$B_{2} \cdot \hat{n} = B_{1} \cdot \hat{n}$$
$$B_{2} \times \hat{n} =\frac{\mu_{2}}{\mu_{1}} B_{1} \times \hat{n},$$
where ##\hat{n}## is a unit radial vector.


Laplace equation:
$$\Delta \phi_{M} = 0,$$
where ##\phi_{M}## is a scalar magnetic potential.

The Attempt at a Solution


I was trying to solve it but I think something's missing here. I mean, in electrostatics I know that some potential ##\phi = 0## on a conducting electrically neutral surface. Should I assume here that there's some current density on the surface of the cylinder?
 

Answers and Replies

  • #2
Charles Link
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For this geometry, there is a well-known demagnetizing factor ## D=\frac{1}{2} ##. It can be solved by Legendre methods that will get the same result, but the easiest way is to use the demagnetizing factor. (The demagnetizing factor allows for the computation of a known uniform "magnetic field" ## H_D ## inside the cylinder that results from the magnetic surface charge ## \sigma_m=M \cdot \hat{n} ##, analogous to the electrostatic problem with the same geometry). ## \\ ## To show you how the demagnetizing factor works, let's assume units where ## B=\mu_o H+M ##. Then ## H_{int}=H_o +H_D ##. ## \\ ## Next ## H_D=-\frac{D M}{\mu_o} ##, and ## M=\mu_o \chi_m H_{int} ## where ## \chi_m=\mu_r-1 ##, with ## B=\mu H=\mu_o \mu_r H ##. ## \\ ## You then solve for ## H_{int} ## and ## M ##. (Basically two equations and two unknowns). You then compute ## B_{int} ##. ## \\ ## Edit: Here is a thread where the demagnetizing factor 1/2 is essentially computed for this problem with a unique method by Griffiths. https://www.physicsforums.com/threa...ormly-polarized-cylinder.941830/#post-5956930 ## \\ ## The alternative is to go the complete Legendre route, but it can be somewhat painstaking. ## \\ ## It might also be of interest that for a spherical geometry ## D=\frac{1}{3} ##, and for a flat disc ## D=1 ##.
 
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