# Energy density in magnetic field

If an electric field $$\mathbf{E}$$ exists at a point in free space then the energy density in that point is $$\frac{1}{2} \epsilon_0 |\mathbf{E}|^2$$.

When a field with same magnitude $$|\mathbf{E}|$$ exists in a material with relative permettivity $$k$$, the energy density becomes $$\frac{1}{2} \epsilon_0 k |\mathbf{E}|^2$$, i.e. it increases $$k$$ times.

A similar analysis for energy density in magnetic field $$\frac{1}{2 \mu_0} |\mathbf{B}|^2$$ states that: if the field $$\mathbf{B}$$ exists in a material with relative permeability $$k$$ (consider a ferromagnetic material, for example) then $$\mu = k \mu_0$$ and thus the energy density decreases $$k$$ times (?). But this does not seem right because inductors with ferromagnetic cores store more energy.

Inside a dielectric, an externally-applied electric field is reduced. Therefore, a higher energy density is required to generate a given electric field inside the dielectric. This means that for a given electric field inside a dielectric, the associated energy density will be larger than in the vacuum; intuitively, you have to "work harder" to produce an electric field in a dielectric, because a dielectric opposes the applied fields.

Inside a ferromagnet, an externally-applied magnetic field is amplified. Therefore, a lower energy density is required to generate a given magnetic field inside the ferromagnet. This means that for a given magnetic field inside a ferromagnet, the associated energy density will be smaller than in the vacuum; intuitively, you "don't have to work as hard" to produce a magnetic field in a ferromagnet, because a ferromagnet amplifies the applied fields.

Along the same lines, ferroelectric materials, which have dielectric constants less than 1, amplify fields like ferromagnets, so the energy density of the electric field inside a ferroelectric material will be reduced relative to the vacuum. Likewise, diamagnetic materials, which have relative permeabilities less than 1, reduce fields like dielectrics, so the magnetic field inside a diamagnetic material will be increased relative to the vacuum.