How is the Total Energy of a Magnetic Field in Matter Calculated?

[LayMuon]

The total energy of the magnetic field in the matter is \frac{\mu H^2}{2}, I want to calculated the energy that is being spent as a the work on magnetizing the material, so I need to subtract the energy of the magnetic field itself \frac{B^2}{2} and the dipolar interaction -\vec{M} \cdot \vec{B}, however here is the problem $$ \frac{\mu H^2}{2} - \frac{B^2}{2} = \frac{\mu H^2}{2} - \frac{(\mu H)^2}{2} < 0 $$ for \mu &gt; &gt; 1.

Why the energy of magnetic field itself is given by \frac{ H^2}{2} and not by \frac{ B^2}{2}?

 

[DrZoidberg]

That equation is for the energy density of the field, not the total energy.
And it doesn't matter if you write \frac{\mu H^2}{2} or \frac{B^2}{2\mu} because B = \mu H

 

[LayMuon]

That all was implied.

The question is why should we take the self energy density of the magnetic field as H^2/2 and not B^2/2, unlike the electric field where it is E^2/2 and not D^2/2.

 

[DrZoidberg]

Why do you keep writing \frac{H^2}{2} instead of \frac{\mu H^2}{2}?
Anyway, That's due to the way H, B and \mu are defined. And because matter often interacts with magnetic fields in a way that's opposite to how it interacts with electric fields.
If you use different definitions, the equations look different.
e.g. in the cgs system the equations are \frac{B^2}{8\pi} and \frac{E^2}{8\pi}