Magnetic field energy inside a linear dielectric

[sudipmaity]

Homework Statement

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 =μ_ο Ho where μ_ο is permeability of the medium in which the sphere is placed.

B_internal=(3μBo z^)/( μ+2 μ_ο)

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

All these answers matched with standard electrodynamics textbooks.But I couldn't 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-μ_ο/μμ_ο)BBo dV.
U = ΔW
U = 1/2 ₀ ∫^a [ B_internal Bo z^]( μ -μ_ο /μμ_ο) 4πr² dr.
Finally U = 2π a³Bo² (μ-μ_ο/2μ_ο+μ)/μ_ο
Change in magnetic field energy must be stored within the sphere as potential energy given by U.

IS MY EXPRESSION FOR U CORRECT??

Homework Equations

The Attempt at a Solution

 

[mark726]

Your expression for the change in magnetic field energy seems to be correct, but there is an error in your final expression for U. The correct expression should be:

U = (4π/3) a^3 Bo^2 (μ-μο)/(2μο+μ)

Also, it is worth noting that this expression for U represents the energy stored in the magnetic field outside of the sphere. The energy stored inside the sphere would be given by:

U' = (4π/3) a^3 Bo^2 (μ-μο)/(μ+2μο)

This is because the energy stored inside the sphere would be equal to the change in energy of the external field, which would be equal to the energy stored in the external field minus the energy stored in the sphere itself. So, the total energy stored would be the sum of U and U'.

Hope this helps!