The current in a coil of self-inductance 2.0 H is increasing according to I = 2 sin(t²) A. The amount of energy spent during the period when the current changes from 0 to 2 A is _________ J.

CONCEPT:

•  The self-inductance for e.m.f of the coil is written as;

            E = L 

            Here we have "I" as current and L is inductance.

• The current is the ratio of change in charge per unit time we have;

           I = 

          Here q as the charge and "I" as the current.

CALCULATION:

Given: I =2sin(t²) A,  Self-inductance of the coil, L = 2.0 H

I = 2sin(t²)   ----(1)

Let us suppose the current is 2 A we have;

2 = 2 sin(t²)

⇒ 1 = sin (t²)

⇒ t² = sin^-1(1)

⇒ t² = 

⇒ t = 

As we have the self-induced for e.m.f we have;

E = L 

The change in the work done is written as;

⇒ W =  L  dq

The current is written as;

I = 

⇒ dq = Idt

Therefore, dW =  L  Idt

dW = LIdi

Now, Integrating above equation we have;

⇒ W = 

⇒ W = 

⇒ W = 8 

Now, by using sin 2t = 2sintcost, therefore;

⇒ W = 4     ----(1)

Now, on putting 2t² = x

⇒ 4t dt = dx

⇒ dt = 

and x =  and x =0

Therefore equation (1) we have;

W = 4L

⇒ W = L 

⇒ W = L 

⇒ W = L(-0+2)

⇒ W = 2 J

Hence, the amount of energy is 2 J