The circuit shown in the figure consists of a battery of emf - - 10 V a capacitor of capacitance C - 1-0 -F and three resistors of values R1- 2 - - R2 - 2 - - R3 - 1 - - Initially- the capacitor is completely uncharged and the switch S is open- The switch S is closed at t -0- Then which of the following statements is-are correct-

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Inductors in Circuits

The circuit s...

Question

The circuit shown in the figure consists of a battery of emf $E = 10 V$ a capacitor of capacitance $C = 1.0 μ F$ and three resistors of values $R_{1} = 2 Ω, R_{2} = 2 Ω & R_{3} = 1 Ω .$ Initially, the capacitor is completely uncharged and the switch $S$ is open. The switch $S$ is closed at $t = 0$ . Then which of the following statements is/are correct?

The current through resistor

R_{3}

at the moment the switch closed is zero

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The current through resistor

R_{3}

a long time after the switch closed is

5 A

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The ratio of current through

R_{1} & R_{2}

is always constant

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The maximum charge on the capacitor during the operation is

5 μ C

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Solution

The correct options are
A The current through resistor $R_{3}$ at the moment the switch closed is zero
B The current through resistor $R_{3}$ a long time after the switch closed is $5 A$
C The ratio of current through $R_{1} & R_{2}$ is always constant
D The maximum charge on the capacitor during the operation is $5 μ C$
Given,
$E = 10 V; C = 1.0 μ F R_{1} = 2 Ω; R_{2} = 2 Ω; R_{3} = 1 Ω$

Just after closing the switch, capacitor will act as wire. Thus, current through the resistor $R_{3}$ is zero.

After closing the switch at $t = \infty,$ the capacitor will act as an open circuit.

$∴$ Current through the battery is,

$I = \frac{V}{R_{3} + \frac{R_{1} R_{2}}{R_{1} + R_{2}}} = \frac{10}{1 + \frac{2 \times 2}{2 + 2}} = 5 A$

$∴ V_{A B} = i R_{3} = 5 V$

The maximum charge on the capacitor is,

$Q = C V = 1.0 μ F \times 5 V = 5 μ C$

Since, $R_{1} & R_{2}$ are in parallel,

$I_{1} R_{1} = I_{2} R_{2}$

$\frac{I_{1}}{I_{2}} = \frac{R_{2}}{R_{1}} = constant$

Hence, all four options are correct.

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