Energy transfers in LR circuits

The gravitational potential energy of a mass changes as the mass moves through a gravitational field. 

The electric potential energy of charge carriers changes as carriers move through a potential difference in a circuit element. 

Consider an inductor, of ohmic resistance r and inductance L. In addition to the Joule heating in the inductor at the rate Pr = ri2, there is also an electric potential energy UL due the self-induced emf ℰ. 

If the current is increasing in the inductor, the self-induced emf opposing the current is negative, hence the carriers lose potential energy. 

If the current is decreasing in the inductor, the self-induced emf opposing the current is positive, hence the carriers gain potential energy. 

The potential energy lost by the charge carriers when the current is increasing is maintained within or stored in the inductor due to the change in the potential difference between the conductor. 

The potential energy gained by the charge carriers when the current is decreasing is returned or restored or recovered by the inductor due to the change in the potential difference between the conductor. 

Each time the potential energy if transformed to and from a stored energy in the inductor, called the magnetic energy of the inductor. 

The power P, or rate of energy used or transformed in the inductor is P = dUL/dt, where dUL is the infinitesimal electric potential energy change stored in the inductor. This change dU corresponds to the change di in current. 

We have P = dUL/dt = ℰi = i L (di/dt) 

P = Li(di/dt) 

Since dUL/dt = i L (di/dt), or 

dUL = L i di, we integrate to obtain 

UL(t) = (1/2) L i2(t) + const. 

At t = 0, UL(t) = 0. Hence const = 0. 

Therefore the energy stored in the inductor is: 

UL = (1/2) L i2 

Energy stored in an inductor: 

UL = (1/2) L i2