The current in a coil of self inductance L = 2 . 0   H is increasing according to the law i = 2 sin t 2 . Find the amount of energy spent (in J ) during the period when the current changes from 0 to 2   A .

Step-by-step solution

Let the current be 2   A at t = τ Then 2 = 2 s i n τ 2 ⇒ τ = π / 2 when the instantaneous current is i , the self-induced emf is L d i d t . If d q amount of charge is displaced in time d t , then elementary work done = L d i d t d q = L d i d t i d t =   L i d i ∴   W = ∫ 0 t   L i d i = ∫ 0 t L 2 s i n t 2 d 2 s i n t 2 ⇒ W = ∫ 0 t 8 L s i n t 2 c o s t 2 t d t = 4 L ∫ 0 t s i n 2 t 2 d t Let θ = 2 t 2 ⇒ d θ = 4 t d