In the circuit shown, find the value of resistance $R$ in terms of inductance $L$ and capacitance $C$ such that the current through the cell remains constant forever after the switch $S$ is closed .

R = \sqrt{\frac{L}{C}}

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R = \sqrt{\frac{2 L}{C}}

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R = \sqrt{\frac{L}{2 C}}

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Not possible for current to be same throughout the time for any value of

R

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Solution

The correct option is A $R = \sqrt{\frac{L}{C}}$
Cell is connected in parallel to $R - L$ path and $R - C$ path.
Current through inductor and capacitor at time $t$ after the switch is closed is,
$i_{L} = \frac{E}{R} ⎡ ⎢ ⎣ 1 - e^{- \frac{R t}{L}} ⎤ ⎥ ⎦$
and $i_{C} = \frac{E}{R} ⎡ ⎢ ⎣ e^{- \frac{t}{R C}} ⎤ ⎥ ⎦$
$\Rightarrow i = i_{C} + i_{L} = \frac{E}{R} ⎡ ⎢ ⎣ 1 - e^{- \frac{R t}{L}} + e^{- \frac{t}{R C}} ⎤ ⎥ ⎦$
This will be time independent only if
$e^{- \frac{R t}{L}} = e^{- \frac{t}{R C}}$
$\Rightarrow \frac{R}{L} = \frac{1}{R C}$
$\Rightarrow R = \sqrt{\frac{L}{C}}$

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In the circuit shown, find the value of resistance R in terms of inductance L and capacitance C such that the current through the cell remains constant forever after the switch S is closed .

In the circuit shown, find the value of resistance $R$ in terms of inductance $L$ and capacitance $C$ such that the current through the cell remains constant forever after the switch $S$ is closed .