(A) In an $LCR$ circuit, the charge on the capacitor at any time $t$ is given by $Q(t) = Q_0 e^{-Rt/2L} \cos(\omega' t + \phi)$.The maximum charge on

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(A) In an $LCR$ circuit, the charge on the capacitor at any time $t$ is given by $Q(t) = Q_0 e^{-Rt/2L} \cos(\omega' t + \phi)$.The maximum charge on the capacitor at any cycle is given by $Q_{max}(t) = Q_0 e^{-Rt/2L}$.Squaring both sides, we get $Q^2_{max}(t) = Q^2_0 e^{-Rt/L}$.This equation represents an exponential decay of the square of the maximum charge with time.The rate of decay depends on the factor $R/L$. Since $L_1 > L_2$, the decay constant $R/L_1$ is smaller than $R/L_2$.Therefore, the charge in the circuit with $L_2$ decays faster than the circuit with $L_1$.Thus, the graph for $L_1$ will show a slower decay compared to the graph for $L_2$.

Class 12 Physics Alternating Current Mix Examples-Alternating Current

Difficult

An $LCR$ circuit is equivalent to a damped oscillator. $A$ capacitor is charged to a charge $Q_0$ as shown and then connected to $L$ and $R$. If a student plots graphs of the square of the maximum charge on the capacitor $(Q^2_{max})$ versus time for two different values of inductors $L_1$ and $L_2$ $(L_1 > L_2)$, which of the following graphs will correctly represent it? (The diagram is schematic and not drawn to scale.)

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Alternating Current

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Mix Examples-Alternating Current

$A$ coil,a capacitor,and an $AC$ source of $rms$ voltage $24 \,V$ are connected in series. By varying the frequency of the source,a maximum $rms$ current of $6 \,A$ is observed. If the coil is connected to a battery of $emf$ $12 \,V$ and internal resistance $4 \,\Omega$,then the current through it in the steady state is......$A$

Medium

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In an $LCR$ circuit $R = 100 \ \Omega$. When capacitance $C$ is removed,the current lags behind the voltage by $\pi /3$. When inductance $L$ is removed,the current leads the voltage by $\pi /3$. The impedance of the circuit is......$\Omega$.

Medium

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Match List-$I$ with List-$II$:

List-$I$ List-$II$

$A$. $AC$ generator $I$. Detects the presence of current in the circuit

$B$. Galvanometer $II$. Converts mechanical energy into electrical energy

$C$. Transformer $III$. Works on the principle of resonance in $AC$ circuit

$D$. Metal detector $IV$. Changes an alternating voltage for smaller or greater value

Choose the correct answer from the options given below:

List-$I$	List-$II$
$A$. $AC$ generator	$I$. Detects the presence of current in the circuit
$B$. Galvanometer	$II$. Converts mechanical energy into electrical energy
$C$. Transformer	$III$. Works on the principle of resonance in $AC$ circuit
$D$. Metal detector	$IV$. Changes an alternating voltage for smaller or greater value

MediumJEE MAIN 2022

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In the given $AC$ circuit,when switch $S$ is at position $1$,the source $emf$ leads the current by $\pi / 6$. Now,if the switch is at position $2$,then

Difficult

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At a particular angular frequency,the reactance of a capacitor and that of an inductor are the same. If the angular frequency is doubled,the ratio of the reactance of the capacitor to that of the inductor will be:

DifficultMHT CET 2023

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In an $LCR$ circuit $R = 100 \ \Omega$. When capacitance $C$ is removed,the current lags behind the voltage by $\pi /3$. When inductance $L$ is removed,the current leads the voltage by $\pi /3$. The impedance of the circuit is......$\Omega$.

In the given $AC$ circuit,when switch $S$ is at position $1$,the source $emf$ leads the current by $\pi / 6$. Now,if the switch is at position $2$,then

At a particular angular frequency,the reactance of a capacitor and that of an inductor are the same. If the angular frequency is doubled,the ratio of the reactance of the capacitor to that of the inductor will be:

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