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Homework Help: Magnetic field energy inside a linear dielectric

  1. Oct 3, 2013 #1
    1. The problem statement, all variables and given/known data
    A Sphere made of a linear magnetic of radius a and permeability μ is placed in an external uniform magnetic field Bo.
    1) Assuming Laplace equation solutions find magnetic field at an internal and external poinnt.
    2) Find the demagnetization factor of the sphere.
    I had no trouble finding and verifying part 1 and 2 from standard books.I did find part 3 . But not being able to verify it.This is a University exam question.
    Here are the answers that I found out: I assumed Bo =μο Ho where μο is permeability of the medium in which the sphere is placed.

    Binternal=(3μBo z^)/( μ+2 μο)

    Bexternal= Bo z^+ (Bo) (a/r)^3 (μ-μο/2μο+μ)( r^ cosθ + θ^ sin θ)
    The demagnetizing factor is 4π/3.

    All these answers matched with standard electrodynamics textbooks.But I couldnt find answer to the 3rd part.Here is how I am doing it.
    I referred to Pg 166 and 214 of J.D.Jackson's book.
    Before the sphere was introduced the energy in the field was:
    Wo = 1/2 ∫ (BoHo) dV
    After the sphere is introduced the energy changes to
    Wi = 1/2 ∫ (BH) dV
    So the change in energy:
    ΔW = 1/2∫( BoHo -BH) dV = 1/2∫ ( μ-μο/μμο)BBo dV.
    U = ΔW
    U = 1/2 0a [ Binternal Bo z^]( μ -μο /μμο) 4πr2 dr.
    Finally U = 2π a3Bo2 (μ-μο/2μο+μ)/μο
    Change in magnetic field energy must be stored within the sphere as potential energy given by U.


    2. Relevant equations

    3. The attempt at a solution
    Last edited: Oct 3, 2013
  2. jcsd
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