Power, Power Factor & Wattless Current

In a series LCR circuit alternating e.m.f. and current are given by the equations $V = V_0\sin(\omega t)$ and $I = I_0\sin\left(\omega t + \frac{\pi}{3}\right)$ respectively. The average power dissipated in the circuit over one cycle of a.c. is $(\cos 60^\circ = 0.5)$

In series LCR circuit, resistance is $18\ \Omega$ and impedance $33\ \Omega$. An r.m.s. Voltage of $220$ V is applied across the circuit. The true power consumed in a c. circuit is

In an AC circuit $E = 50\sin(500t),\ I = 600\sin\left(500t + \frac{\pi}{3}\right)\text{ mA}$. What is the power dissipated in the circuit? [$\cos 60^\circ = 0.5$]

In an a.c. circuit, if the power factor changes from $0.5$ to $0.25$ then the increase in impedance is

In an a.c. circuit containing L, C and R in series, the ratio of apparent power to the true power is (Z= impedance of the circuit and R= resistance)

A LCR series circuit driven with $E_{rms} = 90\text{ V}$ at frequency $f_{\text{d}} = 30\text{ Hz}$ has resistance $R = 80\ \Omega$, an inductance with inductive reactance $X_L = 20.0\ \Omega$ and capacitance with capacitive reactance $X_C = 80.0\ \Omega$. The power factor of the circuit is ______.

An inductor of inductance $L$ and a resistor of resistance $R$ are joined in series and connected to an a.c. source of frequency $\omega$. The power dissipated in the circuit is

An e.m.f. $e = E_0 \cos\omega t$ is applied to a circuit containing L, C and R in series where $X_L = 4R$ and $X_C = R$. The average power dissipated in the circuit is

In series LCR circuit $R = 10\ \Omega$ and impedance is $30\ \Omega$. An r.m.s. voltage $210$ V, is applied across the circuit. The true power consumed in AC circuit is

An inductor of reactance $100\ \Omega$, a capacitor of reactance $50\ \Omega$, and a resistor of resistance $50\ \Omega$ are connected in series with an AC source of $10$ V, $50$ Hz. Average power dissipated by the circuit is ____

For series LCR circuit, the current is observed to lead the applied voltage. An additional capacitor $C_1$ when joined with capacitor C, makes the power factor of the circuit unity. The capacitor $C_1$ must be connected in

An a.c. source is applied to a series LR circuit with $X_L = 3R$ and power factor is $X_1$. Now a capacitor with $X_c = R$ is added in series to LR circuit and power factor is $X_2$. The ratio $X_1$ to $X_2$ is

A series LCR circuit is connected to an a.c. source of $220$ V, $50$ Hz. The circuit contains a resistance $R = 80\ \Omega$, an inductor of inductive reactance $70\ \Omega$ and capacitor of capacitive reactance $130\ \Omega$. The power factor of the circuit is $\frac{x}{10}$. The value of x is

In a series LR circuit with $X_L = R$. 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:

For the series LCR circuit connected with $220\text{ V}, 50\text{ Hz}$ a.c source as shown in the figure, the power factor is $\frac{\alpha}{10}$. The value of $\alpha$ is ______.

An inductor of 10 mH, capacitor of $0.1\ \mu\text{F}$ and a resistor of $100\ \Omega$ are connected in series across an $a.c$ power supply $220\text{ V}, 70\text{ Hz}$. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is $\sqrt{3}\alpha\ \Omega$. The value of $\alpha$ is ______.

An arc lamp requires a direct current of $10$ A at $90$ V to function. If it is connected to $220$ V (rms), $50$ Hz a.c. supply, the series inductor needed for it to work is nearly

In series LCR circuit R $= 18\ \Omega$ and impedance is $30\ \Omega$. An rms voltage 210 V is applied across the circuit. The true power consumed in AC circuit is nearly

In LCR series circuit, R = 18 $\Omega$ and impedance $33\ \Omega$. An r.m.s. voltage of 220 V is applied across the circuit. The true power consumed in a.c. circuit is

In an a.c. circuit containing $L, C, R$ in series the ratio of true power to apparent power is ( $Z =$ impedance of the circuit and R is the resistance)

If the power factor changes from 0.5 to 0.25 because impedance changes from $Z_1$ to $Z_2$ then $Z_1 = xZ_2$. The value of x is (Resistance remains constant)

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.

Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor $\left(R/\sqrt{R^2 + \omega^2 L^2}\right)$, where $\omega$ is frequency of the supply across resistor $R$ and inductor $L$. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage. In the light of the above statements, choose the most appropriate answer from the options given below:

An e.m.f. $e = E_0 \cos\omega t$ is applied to a circuit containing $L, C$ and $R$ in series where $X_L = 3R$ and $X_C = R$. The average power dissipated in the circuit is

In an $L - R$ circuit, the inductive reactance is equal to $\sqrt{3}$ times the resistance ' $R$ ' of the circuit. An e.m.f. $E = E_0 \sin(\omega t)$ is applied to the circuit. The power consumed in the circuit is

An inductor of reactance $100\ \Omega$, a capacitor of reactance $50\ \Omega$, and a resistor of resistance $50\ \Omega$ are connected in series with an AC source of $10\text{ V}, 50\text{ Hz}$. Average power dissipated by the circuit is ___________ W.

The power factor of a CR circuit is $\frac{1}{\sqrt{2}}$, If the frequency of a.c. signal is halved, then the power factor of the circuit will become

When alternating current is passed through $L - R$ series circuit, the power factor is $\frac{\sqrt{3}}{2}$ and $R = 50\ \Omega$, then the value of L is

$\left[\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \sin\frac{\pi}{6} = \frac{1}{2}, \tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}\right]$

An a.c. source is applied to a series LR circuit with $X_L = 3R$ and power factor is $X_1$. Now a capacitor with $X_c = R$ is added in series to LR circuit and the power factor is $X_2$. The ratio $X_1$ to $X_2$ is

In LCR series circuit, an alternating e.m.f. 'e' and current ' $i$ ' are given by equations $e = 160\sin(100t)$ Volt and $i = 250\sin\left(100t + \frac{\pi}{3}\right)\text{ mA}$. The average power dissipated in the circuit is

In an $AC$ circuit $E = 200\sin(50t)$ volt and $I = 100\sin\left(50t + \frac{\pi}{3}\right)\text{ mA}$. The power dissipated in the circuit is

$\left(\begin{matrix}\sin 30^\circ = \cos 60^\circ = 0.5 \\ \sin 60^\circ = \cos 30^\circ = \sqrt{3}/2\end{matrix}\right)$

In an A.C. circuit, the potential difference ' $V$ ' and current 'I' are given respectively by $V = 100\sin(100t)\text{ V}, I = 100\sin\left(100t + \frac{\pi}{3}\right)\text{ mA}$ The power dissipated in the circuit will be [Given $\rightarrow \cos\frac{\pi}{3} = \frac{1}{2}$ ]

An AC voltage $V = 20\sin 200\pi t$ is applied to a series LCR circuit which drives a current $I = 10\sin\left(200\pi t + \frac{\pi}{3}\right)$. The average power dissipated is:

In an a.c. circuit, voltage and current are given by:

$V = 100\sin(100t)\text{ V}$ and $I = 100\sin\left(100t + \frac{\pi}{3}\right)\text{ mA}$ respectively.

The average power dissipated in one cycle is:

A series LCR circuit with $L = \frac{100}{\pi}\text{ mH}, C = \frac{10^{-3}}{\pi}\text{ F}$ and $R = 10\ \Omega$, is connected across an ac source of $220\text{ V}, 50\text{ Hz}$ supply. The power factor of the circuit would be ________.

Average power associated with an ideal inductor and ideal capacitor over a complete cycle of a.c. is respectively

An e.m.f. $E = E_0 \cos\omega t$ is applied to the $L - R$ circuit. The inductive reactance is equal to the resistance ' $R$ ' of the circuit. The power consumed in the circuit is

An e.m.f. $E = E_0 \cos\omega t$ is applied to circuit containing L and R in series. If $X_L = 2R$, then the power dissipated in the circuit is

A capacitor of reactance $4\sqrt{3}\ \Omega$ and a resistor of resistance $4\ \Omega$ are connected in series with an ac source of peak value $8\sqrt{2}\text{ V}$. The power dissipation in the circuit is __________ W.

Match List I with List II

Choose the correct answer from the options given below:

A series L.R circuit connected with an ac source $E = (25\sin 1000t)\text{ V}$ has a power factor of $\frac{1}{\sqrt{2}}$. If the source of emf is changed to $E = (20\sin 2000t)\text{ V}$, the new power factor of the circuit will be:

In an $AC$ circuit, the current is $i = 5\sin\left(100t - \frac{\pi}{2}\right)\text{ A}$ and voltage is $e = 200\sin(100t)$ volt. Power consumption in the circuit is $(\cos 90^\circ = 0)$

For a purely inductive or a purely capacitive circuit, the power factor is

The maximum power is dissipated for an ac in a/an:

Given below are two statements:

Statement I: Maximum power is dissipated in a circuit containing an inductor, a capacitor and a resistor connected in series with an AC source, when resonance occurs

Statement II: Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage.

In the light of the above statements, choose the correct answer from the options given below:

What will be the phase difference between virtual voltage and virtual current when current in the circuit is wattless?

In a $L\text{-}R$ circuit the inductive reactance is equal to the resistance $R$ in the circuit. An emf $E = E_0 \cos\omega t$ is applied to the circuit. The power consumed in the circuit is

A series LCR circuit is connected to an AC source of 220 V, 50 Hz. The circuit contains a resistance R = 80$\Omega$, an inductor of inductive reactance $X_L = 70\ \Omega$, and a capacitor of capacitive reactance $X_C = 130\ \Omega$. The power factor of circuit is $\frac{x}{10}$. The value of $x$ is:

The power factor of an R-L circuit is $\frac{1}{\sqrt{2}}$. If the frequency of $AC$ is doubled the power factor will now be

In a series LR circuit, $X_L = R$, 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

In a series LR circuit with $X_L = R$, power factor P1. If a capacitor of capacitance C with $X_C = X_L$ is added to the circuit the power factor becomes P2. The ratio of P1 to P2 will be:

If wattless current flows in the AC circuit, then the circuit is:

In an A.C circuit, V & I are given by $V = 250\sin(100t)$ volt and $I = 10\sin\left(100t + \frac{\pi}{3}\right)\text{ A}$ respectively The power dissipated in the circuit is:

In a series $LR$ circuit $X_L = R$ and power factor of the circuit is $P_1$. When capacitor with capacitance $C$ such that $X_L = X_C$ is put in series, the power factor becomes $P_2$. The ratio $\frac{P_1}{P_2}$ is:

To light, a $50\text{ W}, 100\text{ V}$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi\sqrt{x}}\ \mu F$, with $200\text{ V}, 50\text{ Hz AC}$ source. The value of $x$ will be ___________.

An AC source is connected to an inductance of 100 mH, a capacitance of 100 $\mu$F and a resistance of 120 $\Omega$ as shown in figure. The time in which the resistance having a thermal capacity 2 J/$^\circ$C will get heated by 16$^\circ$ C is _____________ s.

In a series LCR circuit, the inductive reactance (XL) is 10$\Omega$ and the capacitive reactance (XC) is 4$\Omega$. The resistance (R) in the circuit is 6$\Omega$. The power factor of the circuit is:

An ac circuit has an inductor and a resistor of resistance R in series, such that XL = 3R. Now, a capacitor is added in series such that XC = 2R. The ratio of new power factor with the old power factor of the circuit is $\sqrt{5}:x$. The value of x is ___________.

Two circuits are shown in the figure (a) & (b). At a frequency of ____________ rad/s the average power dissipated in one cycle will be same in both the circuits.

A series LCR circuit driven by 300 V at a frequency of 50 Hz contains a resistance R = 3 k$\Omega$, an inductor of inductive reactance XL = 250 $\pi\ \Omega$ and an unknown capacitor. The value of capacitance to maximize the average power should be: (Take $\pi$2 = 10)

An LCR circuit contains resistance of 110$\Omega$ and a supply of 220 V at 300 rad/s angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by 45$^\circ$. If on the other hand, only inductor is removed the current leads by 45$^\circ$ with the applied voltage. The rms current flowing in the circuit will be:

A 750 Hz, 20 V (rms) source is connected to a resistance of 100 $\Omega$, an inductance of 0.1803 H and a capacitance of 10 $\mu$F all in series. The time in which the resistance (heat capacity 2 J/oC) will get heated by 10oC. (assume no loss of heat to the surroudnings) is close to:

In a series LR circuit, power of 400W is dissipated from a source of 250 V, 50 Hz. The power factor of the circuit is 0.8. In order to bring the power factor to unity, a capacitor of value C is added in series to the L and R. Taking the value of C as $\left(\frac{n}{3\pi}\right)\mu$F, then value of n is __________.

A series AC circuit containing an inductor (20 mH), a capacitor (120 $\mu$F) and a resistor (60 $\Omega$) is driven by an AC source of 24V/50 Hz. The energy dissipated in the circuit in 60 s is:

An inductor 20 mH, a capacitor 100 $\mu$F and a resistor 50 $\Omega$ are connected in series across a source of emf, V = 10 sin 314 t. The power loss in the circuit is

In an a.c. circuit, the instantaneous e.m.f. and current are given by

e = 100 sin 30 t

i = 20 sin $\left(30t - \frac{\pi}{4}\right)$

In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively

The potential differences across the resistance, capacitance and inductance are 80 V, 40 V and 100 V respectively in an L-C-R circuit. The power factor of this circuit is

An inductor 20 mH, a capacitor 50 $\mu F$ and a resistor 40 $\Omega$ are connected in series across a source of emf V = 10 sin 340t. The power loss in A.C. circuit is

An arc lamp requires a direct current of 10 A at 80 V to function. If it is connected to a 220 V (rms), 50 Hz AC supply, the series inductor needed for it to work is close to:

A resistance 'R' draws power 'P' when connected to an AC source. If an inductance is now placed in series with the resistance, such that the impedance of the circuit becomes 'Z'. the power drawn will be

For the LCR circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor C', when joined with the capacitor C present in the circuit, makes the power factor of the circuit unity. The capacitor C', must have been connected in:

A series LR circuit is connected to an ac source of frequency $\omega$ and the inductive reactance is equal to 2R. A capacitance of capacitive reactance equal to R is added in series with L and R. The ratio of the new power factor to the old one is:

The instantaneous values of alternating current and voltage in a circuit are given as $i = \frac{1}{\sqrt{2}}$ sin (100 $\pi$t) ampere

$e = \frac{1}{\sqrt{2}}\sin\left(100\pi t + \frac{\pi}{3}\right)$ Volt

The average power in watts consumed in the circuit is

In a series $LCR$ circuit $R = 200\ \Omega$ and the voltage and the frequency of the main supply are $220\text{ V}$ and $50\text{ Hz}$ respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $30^\circ.$ On taking out the inductor from the circuit the current leads the voltage by $30^\circ.$ The power dissipated in the $LCR$ circuit is

Power dissipated in an LCR series circuit connected to an A.C. source of emf $\varepsilon$ is

In an a.c. circuit the e.m.f. ($\varepsilon$) and the current

(i) at any instant are given respectively by

$\varepsilon$ = E0sin$\omega$t, $i = I$0sin($\omega$t $-\phi$)

The average power in the circuit over one cycle of a.c. is

In an $\text{a.c.}$ circuit the voltage applied is $E = E_0\sin\omega t.$ The resulting current in the circuit is $I = I_0\sin\left(\omega t - \frac{\pi}{2}\right).$ The power consumption in the circuit is given by

A coil of inductive reactance 31 $\Omega$ has a resistance of 8 $\Omega$. It is placed in series with a condenser of capacitative reactance 25 $\Omega$. The combination is connected to an a.c. source of 110 V. The power factor of the circuit is

A circuit has a resistance of $12\ \text{ohm}$ and an impedance of $15\ \text{ohm}$. The power factor of the circuit will be

For a series LCR circuit the power loss at resonance is

The power factor of $AC$ circuit having resistance $(R)$ and inductance $(L)$ connected in series and an angular velocity $\omega$ is

In an a.c. circuit with phase voltage $V$ and current $I$, the power dissipated is

In an A.C. circuit, the current flowing is $I = 5\sin(100t - \pi/2)$ ampere and the potential difference is $V = 200\sin(100t)$ volts. The power consumption is equal to