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Prove. Given a series circuit consisting of a battery of emf V, a resistanceless inductor L and a noninductive resistor R as shown in Fig. 9, the current i at time t is given by

where I_{s} is the steady state current given by I_{s} =
V/R and the switch closing occurs at time t = 0.

Proof. Let us start at point a and go around the circuit in the counterclockwise direction applying Kirchhoff’s laws noting that since the current in increasing, the direction of the emf of the inductor is opposite to the current direction. We get

This is a differential equation whose solution is given by

as can be verified by differentiation and substitution into 2). Since I_{s} = V/R, 3) becomes

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