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

Explanation

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 this opposition by some external agency. This work done is stored as stored as manetic potential energy.
Let us assume that electrical resistance of the inductor is negligible and inductor effect alone is considered. The induced emf e at any intant t is
`e=-L(di)/(dt) " "...(1)`
Let dW be work done on moving a charge dq in a time dt against the opposition, then
`dW=-edq`
`=-eidt" "thereforedq=idt`
Substiuting for e from equation (1)
`-(-L(di)/(dt))`
`dW=Lidi`
Total work done in establishing the current i is
`w=intdW=int_(0)^(i)Lidi=L[i^(2)/(2)]_(0)^(i)`
`W=1/2Li^(2)`
This worked done is stored as magnetic potential energy.
`thereforeu_(B)=1/2Li^(2)" "...(1)`
The energy density is the energy stored per unit volume of the space and is given by
`u_(B)=(U_(B))/(Al)`
`therefore` volume of the solenoid =Al
`u_(B)=(Li^(2))/(2Al)`
`=((mu_(0)n^(2)Al)i^(2))/(2Al) thereforeL=mu_(0)n^(2)Al`
`=(mu_(0)n^(2)i^(2))/(2)" "thereforeB=mu_(0)ni`
`u_(B)=(B^(2))/(2mu_(0))`