First Order Circuit

RL Circuit

Energy Conservation

As current increases, work done by the battery is not the same as the energy dissipated by the resistor. The remaining energy,

$L/2I^{2}$

is stored in the form of magnetic field inside the inductor. Substituting inductance and factoring out volume, we eventually get energy density

${\frac {B^{2}}{2\mu _{0}}}$