An inductor of inductance L carries an electric current i. How much energy is stored while establishing the current in it ?

Explanation

Energy stoted in an inductor : Whenever a current is established in the circuit, the inductance opposes the growth of the current . In order to establish a current in the circuit , work is done against this opposition by some exteranl agency. This work done is stored as magnetic potential energy. Let us assume that electrical resistance of the inductor is negligible and inductor effect alone is considered. The induced emf `varepsilon` at any instant t is
`varepsilon =-L (di)/(dt)` ........(1)
Let dW be work done in moving a charge dq in a time dt against the opposition, then
dW =`-varepsilondq =-varepsilonidi` `[because dq =idt]`
substituting for `varepsilon` value from equation (1)
`=-(-L(di)/(dt))idt`
dW =Lidi ...........(2)
Total work done in establishing the current i is
`W=int dW = int _(0)^(i) Lidi =L [(i^2)/2]_0^i`
`W=1/2 Li^2` ...........(3)
This work done is stored as magnetic potential eneergy
`therefore U_B =1/2 Li^2`