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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 Is is the steady state current given by Is = 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 Is = V/R, 3) becomes