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Question: Average energy stored in a pure inductance L when a current i flows through it, is...

Average energy stored in a pure inductance L when a current i flows through it, is

A

Li2L i ^ { 2 }

B

2Li22 L i ^ { 2 }

C

Li24\frac { L i ^ { 2 } } { 4 }

D

Li22\frac { L i ^ { 2 } } { 2 }

Answer

Li22\frac { L i ^ { 2 } } { 2 }

Explanation

Solution

As we know e=−dϕdt=−Ldidte = - \frac { d \phi } { d t } = - L \frac { d i } { d t }

Work done against back e.m.f. e in time dt and current i is

dW=−eidt=Ldidtidt=Lidid W = - e i d t = L \frac { d i } { d t } i d t = L i d i ⇒W=L∫0iidi=12Li2\Rightarrow W = L \int _ { 0 } ^ { i } i d i = \frac { 1 } { 2 } L i ^ { 2 }