Magnetic field energy inside a linear dielectric

1. Oct 3, 2013

sudipmaity

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.
3) CALCULATE THE MAGNETIC FIELD ENERGY STORED WITHIN 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.

IS MY EXPRESSION FOR U CORRECT??

2. Relevant equations

3. The attempt at a solution

Last edited: Oct 3, 2013