First slide

Time -Varying current in L-R circuit

Question

At t =0, an inductor of zero resistance is joined to a cell of emf through a resistance. The current increases with a time constant $τ$ . After what time will the potential difference across the coil be equal to that across the resistance?

Moderate

A

T
B

T 1n 2
C

T (1-ln2)
D

T /1n2

Solution

$\begin{array}{l} i = i_{o} (1 - e^{- \frac{R}{2} t}) \\ \frac{di}{dt} = i_{o} \frac{R}{2} e^{- \frac{R}{2} t} \\ L \frac{di}{dt} = i_{o} {Re}^{- \frac{R}{2} t} = \frac{i_{o} R}{2} \\ e^{- \frac{R}{2} t} = 2^{- 1} \\ e^{\frac{R}{2} t} = 2 \\ \ln 2 = \frac{R}{L} t \\ t = \frac{L}{R} \ln (2) = τln (2) \end{array}$

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