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How to calculate the energy in a metal object subject to an external magnetic field?

  1. Mar 8, 2008 #1

    I'd like to know if there is some way of computing or deriving the energy stored in a metallic object subject to a magnetic field from external sources?

    Say, an iron cylinder or a sphere in a magnetic field (that is uniform or non-uniform). What is the energy in the cylinder because of this external magnetic field? Is there some formula that I can use or some concept or site that I can look at to understand this? Been stuck with this for a while and my results don't look right.

    Appreciate any help in advance.
  2. jcsd
  3. Mar 8, 2008 #2
    Btw, I'd appreciate a strong mathematical description (link, theory, idea), etc. Am a grad student, so the math itself isn't the problem. Just the concept, taking into account interface conditions, etc. I assume that's how it's done in PF? Suggestions are always welcome.
  4. Mar 8, 2008 #3


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    The energy depends on the geometry of the object, among other things.
    A soft iron sphere in an originally uniform magnetic field B_0 is probably the easiest case.
    The magnetization in the sphere is uniform, given by M=3B_0/4pi (in Gaussian units).
    This is found in the same way as for a dielectric sphere in a uniform electric field.
    The magnetic energy is given by [tex]U=(1/8\pi)\int B^2d^r[/tex],
    for which I get U=B_0^2R^3/4pi.
  5. Mar 9, 2008 #4
    Thanks for that Pam,

    But what is the basic relation that I derive this from? I'd like to do it for a cylinder and for some other shapes too. Wouldn't the field depend on the angle that each small area makes with the external field?

    And is that integrand B^2*d^r? What's that notation?
  6. Mar 9, 2008 #5


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    I was a bit careless, The integral should be
    [tex]U=(1/8\pi)\int {\vec B}\cdot{\vec H}d^3r[/tex].
    However, since H~0 inside a high mu sphere, the integral is for all space outside the sphere where B=H, so the B^2 is correct here.
    The problem for a soft iron sphere is just the same as for a dielectric sphere in an electric field, which is done in most EM books. Then, the B field for r>R is just that of a dipole.
    The energy is very shape dependent. For a long narrow cylinder aligned with the B_0 field, or for a disk, the calculation is also fairly simple.
    The procedure is:
    Use the standard electrostatic methods, just letting E-->H, D-->B, P-->M, epsilon-->mu
    to find B, H, and M. Then do the integral over B.H.
    For more complicated, shapes, the first step becomes very complicated.
  7. Mar 12, 2008 #6
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