(B) The power factor of an $LR$ circuit is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X_L^2}}$.Given $X_L = R$,we have $P_1 = \frac{R}{\

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(B) The power factor of an $LR$ circuit is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X_L^2}}$.Given $X_L = R$,we have $P_1 = \frac{R}{\sqrt{R^2 + R^2}} = \frac{R}{\sqrt{2R^2}} = \frac{1}{\sqrt{2}}$.When a capacitor $C$ is added such that $X_C = X_L$,the circuit becomes a series $LCR$ circuit at resonance.The impedance of the $LCR$ circuit is $Z = \sqrt{R^2 + (X_L - X_C)^2}$.Since $X_L = X_C$,the impedance $Z = \sqrt{R^2 + 0} = R$.The new power factor is $P_2 = \frac{R}{Z} = \frac{R}{R} = 1$.Therefore,the ratio $P_1 : P_2 = \frac{1}{\sqrt{2}} : 1 = 1 : \sqrt{2}$.

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

Medium

In a series $LR$ circuit,$X_L=R$,the power factor is $P_1$. If a capacitor of capacitance $C$ with $X_C=X_L$ is added to the circuit,the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be

MHT CET 2023 All MHT CET

A
$1: 3$
B
$1: \sqrt{2}$
C
$1: 1$
D
$1: 2$

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Alternating Current

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RL, RC and LC AC Circuits

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$A$ resistor of $500 \Omega$ and an inductor of $0.5 \ H$ are connected in series with an $AC$ source given by $V = 100 \sqrt{2} \sin(1000 t)$. The power factor of the combination is:

MediumKCET 2012

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An inductive coil has a resistance of $100 \Omega$. When an a.c. signal of frequency $1000 \ Hz$ is applied to the coil,the voltage leads the current by $45^{\circ}$. The inductance of the coil is

EasyMHT CET 2021

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$A$ series $R-C$ combination is connected to an $AC$ voltage of angular frequency $\omega = 500 \ rad/s$. If the impedance of the $R-C$ circuit is $R\sqrt{1.25}$,the time constant (in $ms$) of the circuit is:

DifficultIIT 2011

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$A$ series $LCR$ circuit consists of $R = 80\,\Omega$,$X_{L} = 100\,\Omega$,and $X_{C} = 40\,\Omega$. The input voltage is $V = 2500 \cos(100\pi t)\,V$. The amplitude of current in the circuit is $................A$.

MediumJEE MAIN 2023

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

List-$I$ List-$II$

$(a)$ Phase difference between current and voltage in a purely resistive $AC$ circuit $(i)$ $\frac{\pi}{2}$; current leads voltage

$(b)$ Phase difference between current and voltage in a pure inductive $AC$ circuit $(ii)$ zero

$(c)$ Phase difference between current and voltage in a pure capacitive $AC$ circuit $(iii)$ $\frac{\pi}{2}$; current lags voltage

$(d)$ Phase difference between current and voltage in an $LCR$ series circuit $(iv)$ $\tan^{-1}\left(\frac{X_C - X_L}{R}\right)$

Choose the most appropriate answer from the options given below:

List-$I$	List-$II$
$(a)$ Phase difference between current and voltage in a purely resistive $AC$ circuit	$(i)$ $\frac{\pi}{2}$; current leads voltage
$(b)$ Phase difference between current and voltage in a pure inductive $AC$ circuit	$(ii)$ zero
$(c)$ Phase difference between current and voltage in a pure capacitive $AC$ circuit	$(iii)$ $\frac{\pi}{2}$; current lags voltage
$(d)$ Phase difference between current and voltage in an $LCR$ series circuit	$(iv)$ $\tan^{-1}\left(\frac{X_C - X_L}{R}\right)$

MediumJEE MAIN 2021

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In a series $LR$ circuit,$X_L=R$,the power factor is $P_1$. If a capacitor of capacitance $C$ with $X_C=X_L$ is added to the circuit,the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be

Similar Questions

$A$ resistor of $500 \Omega$ and an inductor of $0.5 \ H$ are connected in series with an $AC$ source given by $V = 100 \sqrt{2} \sin(1000 t)$. The power factor of the combination is:

An inductive coil has a resistance of $100 \Omega$. When an a.c. signal of frequency $1000 \ Hz$ is applied to the coil,the voltage leads the current by $45^{\circ}$. The inductance of the coil is

$A$ series $R-C$ combination is connected to an $AC$ voltage of angular frequency $\omega = 500 \ rad/s$. If the impedance of the $R-C$ circuit is $R\sqrt{1.25}$,the time constant (in $ms$) of the circuit is:

$A$ series $LCR$ circuit consists of $R = 80\,\Omega$,$X_{L} = 100\,\Omega$,and $X_{C} = 40\,\Omega$. The input voltage is $V = 2500 \cos(100\pi t)\,V$. The amplitude of current in the circuit is $................A$.

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