(D) The initial impedance is $Z = \sqrt{R^2 + X_c^2}$,where $X_c = \frac{1}{\omega C}$. The initial current is $I = \frac{V}{Z}$.When the frequency ch

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(D) The initial impedance is $Z = \sqrt{R^2 + X_c^2}$,where $X_c = \frac{1}{\omega C}$. The initial current is $I = \frac{V}{Z}$.When the frequency changes to $\omega' = \frac{\omega}{3}$,the new reactance is $X_c' = \frac{1}{\omega' C} = \frac{1}{(\omega/3)C} = 3X_c$.The new impedance is $Z' = \sqrt{R^2 + (3X_c)^2} = \sqrt{R^2 + 9X_c^2}$.The new current is $I' = \frac{V}{Z'} = \frac{I}{2}$,which implies $Z' = 2Z$.Squaring both sides,$Z'^2 = 4Z^2$,so $R^2 + 9X_c^2 = 4(R^2 + X_c^2)$.$R^2 + 9X_c^2 = 4R^2 + 4X_c^2$.$5X_c^2 = 3R^2$.$\frac{X_c^2}{R^2} = \frac{3}{5}$.Therefore,the ratio $\frac{X_c}{R} = \sqrt{\frac{3}{5}}$.

Class 12 Physics Alternating Current RL, RC and LC AC Circuits

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An $a.c.$ source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current registered is $I$. If the frequency of the source is changed to $\frac{\omega}{3}$ (while maintaining the same voltage),the current in the circuit is found to be halved. The ratio of reactance to resistance at the original frequency $\omega$ is:

MHT CET 2021 All MHT CET

A
$\sqrt{\frac{2}{5}}$
B
$\sqrt{\frac{1}{5}}$
C
$\sqrt{\frac{4}{5}}$
D
$\sqrt{\frac{3}{5}}$

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Alternating Current

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RL, RC and LC AC Circuits

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An a.c. source of $15 \, V, 50 \, Hz$ is connected across an inductor $(L)$ and resistance $(R)$ in series. The $R.M.S.$ current of $0.5 \, A$ flows in the circuit. The phase difference between the applied voltage and current is $\left(\frac{\pi}{3}\right)$ radian. The value of resistance $(R)$ is $\left(\tan 60^{\circ}=\sqrt{3}\right)$. (in $\Omega$)

DifficultMHT CET 2023

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An $A.C.$ source is connected to a series $LCR$ circuit. If the voltage across $R$ is $40 \,V$, the voltage across $L$ is $80 \,V$, and the voltage across $C$ is $40 \,V$, then the e.m.f. '$e$' of the $A.C.$ source is:

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$A$ coil has a resistance of $30\,\Omega$ and an inductive reactance of $20\,\Omega$ at a frequency of $50\,Hz$. If an $AC$ source of $200\,V$ and $100\,Hz$ is connected across the coil,the current in the coil will be.......$A$

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When $100 \, V$ $DC$ is supplied across a solenoid,a current of $1.0 \, A$ flows in it. When $100 \, V$ $AC$ is applied across the same coil,the current drops to $0.5 \, A$. If the frequency of the $AC$ source is $50 \, Hz$,then the impedance and inductance of the solenoid are:

Medium

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An a.c. voltage source $V=V_0 \sin \omega t$ is connected across resistance $R$ and capacitance $C$ in series. It is given that $R=\frac{1}{\omega C}$ and the peak current is $I_0$. If the angular frequency of the voltage source is changed to $\frac{\omega}{\sqrt{3}}$,then the new peak current in the circuit is

MediumMHT CET 2024

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An $A.C.$ source is connected to a series $LCR$ circuit. If the voltage across $R$ is $40 \,V$, the voltage across $L$ is $80 \,V$, and the voltage across $C$ is $40 \,V$, then the e.m.f. '$e$' of the $A.C.$ source is:

$A$ coil has a resistance of $30\,\Omega$ and an inductive reactance of $20\,\Omega$ at a frequency of $50\,Hz$. If an $AC$ source of $200\,V$ and $100\,Hz$ is connected across the coil,the current in the coil will be.......$A$

When $100 \, V$ $DC$ is supplied across a solenoid,a current of $1.0 \, A$ flows in it. When $100 \, V$ $AC$ is applied across the same coil,the current drops to $0.5 \, A$. If the frequency of the $AC$ source is $50 \, Hz$,then the impedance and inductance of the solenoid are:

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