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 $\mu > > 1$.

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 $\frac{E^2}{8\pi}$

 

[sardar]

I would like to clarify that the formula for the total energy of the magnetic field in matter is indeed given by \frac{\mu H^2}{2}, where \mu is the permeability of the material and H is the magnetic field strength. This formula takes into account the contribution of both the external magnetic field (H) and the magnetization of the material (\mu).

The confusion may arise from the fact that the energy of the magnetic field itself is often written as \frac{B^2}{2}, where B is the total magnetic field (including the contribution from the material). This is because the total magnetic field (B) is related to the magnetic field strength (H) and the magnetization of the material (\mu) by the equation B = \mu H. Therefore, using this relationship, we can rewrite the formula for the total energy of the magnetic field as \frac{\mu H^2}{2} = \frac{(\mu H)^2}{2} = \frac{B^2}{2}.

Regarding the issue of negative energy, it is important to note that the energy of the magnetic field is a relative quantity and can only be measured in comparison to a reference point. This means that the energy of the magnetic field can be positive or negative depending on the chosen reference point. In this case, the reference point is taken to be the energy of the magnetic field at zero magnetization. When the magnetization of the material is increased, the energy of the magnetic field also increases, but it remains negative with respect to the reference point. This does not mean that the energy is actually negative, but rather that it is lower than the reference point.

In summary, the formula for the total energy of the magnetic field in matter is correct and takes into account all the relevant contributions. The apparent discrepancy between \frac{\mu H^2}{2} and \frac{B^2}{2} is due to the fact that the latter is often used as a simplified expression, taking into account the relationship between B and H. Additionally, the negative energy is a relative quantity and does not have a physical meaning.