Alternating Current
A square shaped coil of area $70 \mathrm{~cm}^{2}$ having 600 turns rotates in a magnetic field of $0.4 ~\mathrm{wbm}^{-2}$, about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes 500 revolution in a minute, the instantaneous emf when the plane of the coil is inclined at $60^{\circ}$ with the field, will be ____________ V. (Take $\pi=\frac{22}{7}$)
Explanation:
$ \mathrm{B}=0.4 \mathrm{~T} $
$f=\frac{500 \text { revolution }}{60 \text { minute }}=\frac{500}{60} \frac{\text { rev. }}{\mathrm{sec} .}$
Induced emf in rotating coil is given by
$ \begin{aligned} & e=N \omega B A \sin \theta \\\\ & =600 \times 2 \times \frac{22}{7} \times \frac{500}{60} \times 0.4 \times 70 \times 10^{-4} \sin 30^{\circ} \\\\ & =600 \times 2 \times \frac{22}{7} \times \frac{500}{6} \times 0.4 \times 70 \times 10^{-4} \times \frac{1}{2} \\\\ & =44 \text { Volt } \end{aligned} $
A series LCR circuit is connected to an ac source of $220 \mathrm{~V}, 50 \mathrm{~Hz}$. The circuit contain a resistance $\mathrm{R}=100 ~\Omega$ and an inductor of inductive reactance $\mathrm{X}_{\mathrm{L}}=79.6 ~\Omega$. The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be _________ $\mu \mathrm{F}$.
Explanation:
So in LCR circuit power will be maximum at the condition of resonance and in resonance condition
$ \begin{aligned} & \therefore X_{L}=X_{C} \\\\ & 79.6=\frac{1}{2 \pi(50) \times C} \\\\ & C=\frac{1}{79.6 \times 2 \pi(50)} \\\\ & \approx 40 \mu \mathrm{F} \end{aligned} $
voltage is 2500 $\cos (100 \pi \mathrm{t}) \mathrm{V}$. The amplitude of current, in the circuit, is _________ A.
Explanation:
$ \begin{aligned} & \text { So } Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}} \\\\ & =\sqrt{80^{2}+(100-40)^{2}} \\\\ & =100 \Omega \\\\ & i_{0}=\frac{V_{0}}{Z}=\frac{2500}{100} \mathrm{~A}=25 \mathrm{~A} \end{aligned} $
An inductor of $0.5 ~\mathrm{mH}$, a capacitor of $20 ~\mu \mathrm{F}$ and resistance of $20 ~\Omega$ are connected in series with a $220 \mathrm{~V}$ ac source. If the current is in phase with the emf, the amplitude of current of the circuit is $\sqrt{x}$ A. The value of $x$ is ___________
Explanation:
So, $\mathrm{Z}=\mathrm{R}=20 \Omega$
$ \begin{aligned} & \mathrm{i}_{\mathrm{rms}}=\frac{220}{20}=11 \\\\ & \mathrm{i}_{\max }=11 \sqrt{2}=\sqrt{242} \end{aligned} $
$\left(\right.$ Take $\left.\pi=\frac{22}{7}\right)$
Explanation:
$\phi=B.A$
$\phi=\mathrm{BNA}\cos\omega t$
So, $Emf = {{ - d\phi } \over {dt}} = NBA\omega \sin \omega t$
So maximum value of emf is
${E_{\max }} = NBA\omega $
$ = 100 \times 3 \times 14 \times {10^{ - 2}} \times {{360 \times 2\pi } \over {60}} = 1584$
An inductor of inductance 2 $\mathrm{\mu H}$ is connected in series with a resistance, a variable capacitor and an AC source of frequency 7 kHz. The value of capacitance for which maximum current is drawn into the circuit $\frac{1}{x}\mathrm{F}$, where the value of $x$ is ___________.
(Take $\pi=\frac{22}{7}$)
Explanation:
Current drawn is maximum when circuit is in resonance.
$\omega = {1 \over {\sqrt {LC} }}$
$2\pi (7000) = {1 \over {\sqrt {2 \times {{10}^{ - 6}}C} }}$
$ \Rightarrow C = {1 \over {3872}}F$
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 $\mathrm{X_L=70\Omega}$, and a capacitor of capacitive reactance $\mathrm{X_C=130\Omega}$. The power factor of circuit is $\frac{x}{10}$. The value of $x$ is :
Explanation:
So, $x=8$
An LCR series circuit of capacitance 62.5 nF and resistance of 50 $\Omega$, is connected to an A.C. source of frequency 2.0 kHz. For maximum value of amplitude of current in circuit, the value of inductance is __________ mH.
(Take $\pi^2=10$)
Explanation:
$ \begin{aligned} & \therefore X_{L}=X_{C} \\\\ & \omega L=\frac{1}{\omega C} \\\\ & L=\frac{1}{\omega^{2} C} \\\\ & =\frac{1}{\left(2 \pi \times 2 \times 10^{3}\right)^{2} \times 62.5 \times 10^{-9}} \\\\ & =100 ~\mathrm{mH} \end{aligned} $
In the circuit shown in the figure, the ratio of the quality factor and the band width is ___________ s.
Explanation:
Quality factor $Q=\frac{1}{R} \sqrt{\frac{L}{C}}$
So $\frac{Q}{\Delta \omega}=\frac{\frac{1}{R} \sqrt{\frac{L}{C}}}{\frac{R}{L}}$
$ \begin{aligned} & =\frac{L^{\frac{3}{2}}}{R^{2} \sqrt{C}} \\\\ & =\frac{3^{\frac{3}{2}}}{10^{2}\left(27 \times 10^{-6}\right)^{\frac{1}{2}}} \\\\ & =\frac{3 \sqrt{3}}{100\left(3 \sqrt{3} \times 10^{-3}\right)} \\\\ & =10 \end{aligned} $
Given below are two statements:
Statement I : An AC circuit undergoes electrical resonance if it contains either a capacitor or an inductor.
Statement II : An AC circuit containing a pure capacitor or a pure inductor consumes high power due to its non-zero power factor.
In the light of above statements, choose the correct answer form the options given below:
Given below are two statements:
Statement I : When the frequency of an a.c source in a series LCR circuit increases, the current in the circuit first increases, attains a maximum value and then decreases.
Statement II : In a series LCR circuit, the value of power factor at resonance is one.
In the light of given statements, choose the most appropriate answer from the options given below.
As per the given graph, choose the correct representation for curve $\mathrm{A}$ and curve B.
Where $\mathrm{X}_{\mathrm{C}}=$ reactance of pure capacitive circuit connected with A.C. source
$\mathrm{X}_{\mathrm{L}}=$ reactance of pure inductive circuit connected with $\mathrm{A} . \mathrm{C}$. source
R = impedance of pure resistive circuit connected with A.C. source.
$\mathrm{Z}=$ Impedance of the LCR series circuit $\}$
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:
A capacitor of capacitance $150.0 ~\mu \mathrm{F}$ is connected to an alternating source of emf given by $\mathrm{E}=36 \sin (120 \pi \mathrm{t}) \mathrm{V}$. The maximum value of current in the circuit is approximately equal to :
Match List - I with List - II :
| List I | List II | ||
|---|---|---|---|
| A. | AC generator | I. | Presence of both L and C |
| B. | Transformer | II. | Electromagnetic Induction |
| C. | Resonance phenomenon to occur | III. | Quality factor |
| D. | Sharpness of resonance | IV. | Mutual Induction |
Choose the correct answer from the options given below :
If $\mathrm{R}, \mathrm{X}_{\mathrm{L}}$, and $\mathrm{X}_{\mathrm{C}}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :
In a series LR circuit with $\mathrm{X_L=R}$, power factor P1. If a capacitor of capacitance C with $\mathrm{X_C=X_L}$ is added to the circuit the power factor becomes P2. The ratio of P1 to P2 will be :
For the given figures, choose the correct options :
In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes $x$ times its initial resonant frequency $\omega_0$. The value of $x$ is :
A capacitor of capacitance 500 $\mu$F is charged completely using a dc supply of 100 V. It is now connected to an inductor of inductance 50 mH to form an LC circuit. The maximum current in LC circuit will be _______ A.
Explanation:
At steady state charge stored on the capacitor,
${q_{\max }} = CV$
$ = 500 \times {10^{ - 6}} \times 100$
$ = 5 \times {10^{ - 2}}\,C$
Energy stored in the capacitor,
${U_{\max }} = {{q_{\max }^2} \over {2C}}$
Now, when electrostatic energy of capacitor converted to magnetic field energy then all energy of capacitor is transferrd to the inductor.
$\therefore$ Maximum energy stored in the inductor
${U_{L\,\max }} = {1 \over 2}L\,I_{\max }^2$
$\therefore$ ${1 \over 2}L\,I_{\max }^2 = {{q_{\max }^2} \over {2C}}$
$ \Rightarrow {I_{\max }} = {{{q_{\max }}} \over {\sqrt {LC} }}$
$ = {{5 \times {{10}^{ - 2}}} \over {\sqrt {50 \times {{10}^{ - 3}} \times 500 \times {{10}^{ - 6}}} }}$
$ = {{5 \times {{10}^{ - 2}}} \over {5 \times {{10}^{ - 3}}}}$
$ = 10\,A$
The frequencies at which the current amplitude in an LCR series circuit becomes $\frac{1}{\sqrt{2}}$ times its maximum value, are $212\,\mathrm{rad} \,\mathrm{s}^{-1}$ and $232 \,\mathrm{rad} \,\mathrm{s}^{-1}$. The value of resistance in the circuit is $R=5 \,\Omega$. The self inductance in the circuit is __________ $\mathrm{mH}$.
Explanation:
${i \over {{i_{\max }}}} = {1 \over {\sqrt 2 }}$
$ = {{{{{V_0}} \over Z}} \over {{{{V_0}} \over R}}}$
$ \Rightarrow {R \over Z} = {1 \over {\sqrt 2 }}$
and ${1 \over {212C}} - 212L = 232L - {1 \over {232C}}$
so $212L = {1 \over {232C}}$
so ${R \over {\sqrt {{R^2} + {{\left( {232L + {1 \over {232C}}} \right)}^2}} }} = {1 \over {\sqrt 2 }}$
${{{R^2}} \over {{R^2} + {{(20L)}^2}}} = {1 \over 2}$
$400{L^2} = {R^2}$
$L = {5 \over {20}}$
$H = {5 \over {20}} \times 1000$ mH
$= 250$ mH
To light, a $50 \mathrm{~W}, 100 \mathrm{~V}$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi \sqrt{x}} \mu F$, with $200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$ source. The value of $x$ will be ___________.
Explanation:
${X_C} = {1 \over {wc}} = {{\pi \sqrt x } \over {2\pi \times 50 \times 50}} \times {10^6}$
$v_R^2 + v_C^2 = {(200)^2}$
$v_C^2 = {200^2} - {100^2}$
${v_C} = 100\sqrt 3 \,V$
${v_R} = 100\,V$
$P = {{{V^2}} \over R}$
$R = {{100 \times 100} \over {50}} = 200\,\Omega $
${i_{rm}} = {1 \over 2}\,A$
${1 \over 2} \times {x_C} = 100\sqrt 3 \Rightarrow {10^{ - 6}} \times {{\sqrt x } \over {5000}} \times {1 \over 2} = 100\sqrt 3 $
${{{{10}^{ - 6}}\sqrt x } \over {10000 \times 100}} = \sqrt 3 $
$\sqrt x = \sqrt 3 $
$x = 3$
The effective current I in the given circuit at very high frequencies will be ___________ A.
Explanation:
Equivalent circuit will be
$I = {{220} \over 5} = 44\,A$
A series LCR circuit with $R = {{250} \over {11}}\,\Omega $ and ${X_L} = {{70} \over {11}}\,\Omega $ is connected across a 220 V, 50 Hz supply. The value of capacitance needed to maximize the average power of the circuit will be _________ $\mu$F. (Take : $\pi = {{22} \over 7}$)
Explanation:
$ \begin{aligned} &\text { power factor }=\cos \theta=1\\\\ & \therefore \frac{R}{Z}=1 \\\\ &R^{2}=Z^{2} \\\\ &R^{2}=\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^{2}+\mathrm{R}^{2} \\\\ &\mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}} \\\\ &\frac{70}{11}=\frac{1}{100 \pi \times C} \\\\ &\Rightarrow C=\frac{11}{7000 \pi}=500 \times 10^{-6} F=500 \mu F \end{aligned} $
An inductor of 0.5 mH, a capacitor of 200 $\mu$F and a resistor of 2 $\Omega$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the frequency of ac source will be ____________ $\times$ 102 Hz.
Explanation:
Current will be in phase with emf when
$\omega L = {1 \over {\omega C}}$
$ \Rightarrow \omega = {1 \over {\sqrt {LC} }} = {1 \over {\sqrt {5 \times {{10}^{ - 4}} \times 2 \times {{10}^{ - 4}}} }}$
$ \Rightarrow \omega = {{{{10}^4}} \over {\sqrt {10} }}$ rad/s
$ \Rightarrow f = {1 \over {2\pi }} \times {{{{10}^4}} \over {\sqrt {10} }}$ Hz
$\Rightarrow$ f $\simeq$ 500 Hz
In the given circuit, the magnitude of VL and VC are twice that of VR. Given that f = 50 Hz, the inductance of the coil is ${1 \over {K\pi }}$ mH. The value of K is ____________.
Explanation:
${V_L} = 2{V_R}$
So $\omega Li = 2\,Ri$
$ \Rightarrow L = {{2R} \over \omega } = {{2 \times 5} \over {2\pi \times 50}} = {1 \over {10\pi }}H = {{100} \over \pi }H$
So $k = {1 \over {100}} \simeq 0$
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.
Explanation:
L = 100 $\times$ 10$-$3 H
C = 100 $\times$ 10$-$6 F
R = 120 $\Omega$
$\omega$L = 10 $\Omega$
${1 \over {\omega C}} = {1 \over {{{10}^4} \times {{10}^{ - 6}}}} = 100\,\Omega $
$\Rightarrow$ XC $-$ XL = 90 $\Omega$
$ \Rightarrow Z = \sqrt {{{90}^2} + {{120}^2}} = 150\,\Omega $
$ \Rightarrow {I_{rms}} = {{20} \over {150}} = {2 \over {15}}A$
For heat resistance by 16$^\circ$C heat required = 32 J
$ \Rightarrow {\left( {{2 \over {15}}} \right)^2} \times (120) \times t = 32$
$t = {{32 \times 15 \times 15} \over {4 \times 120}} = 15$
A telegraph line of length 100 km has a capacity of 0.01 $\mu$F/km and it carries an alternating current at 0.5 kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is _____________ mH. (if $\pi$ = $\sqrt{10}$)
Explanation:
Total capacitance = 0.01 $\times$ 100 = 1 $\mu$F
$\omega$ = 500 $\times$ 2$\pi$ = 1000$\pi$ rad/s
$\omega L = {1 \over {\omega C}}$
$ \Rightarrow L = {1 \over {{\omega ^2}C}} = {1 \over {{{10}^6}{\pi ^2} \times {{10}^{ - 6}}}} = {1 \over {10}}H$ = 100 mH
A 220 V, 50 Hz AC source is connected to a 25 V, 5 W lamp and an additional resistance R in series (as shown in figure) to run the lamp at its peak brightness, then the value of R (in ohm) will be _____________.
Explanation:
${R_b} = {{{{(25)}^2}} \over 5} = 125\,\Omega $
${I_{rms}} = \sqrt {{5 \over {125}}} = {1 \over 5}A$
$ \Rightarrow {{220} \over {R + 125}} = {1 \over 5}$
$ \Rightarrow R = 1100 - 125$
$ = 975\,\Omega $
A 110 V, 50 Hz, AC source is connected in the circuit (as shown in figure). The current through the resistance 55 $\Omega$, at resonance in the circuit, will be __________ A.
Explanation:
At resonance, $\mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}} \& \mathrm{Z} \rightarrow \infty$
$\therefore \mathrm{Z}_{\text {total circuit }} \rightarrow \infty$
i.e, $\mathrm{I}=0$
In a series LCR circuit, the inductance, capacitance and resistance are L = 100 mH, C = 100 $\mu$F and R = 10 $\Omega$ respectively. They are connected to an AC source of voltage 220 V and frequency of 50 Hz. The approximate value of current in the circuit will be ___________ A.
Explanation:
$Z = \sqrt {{R^2} + {{({x_L} + {x_C})}^2}} $
$ = \sqrt {{{10}^2} + {{\left[ {10\pi - {{100} \over \pi }} \right]}^2}} \,\Omega $
$ \simeq 10\,\Omega $
$\Rightarrow$ Current $ = {{220} \over {10}}$ A = 22 A
As shown in the figure an inductor of inductance 200 mH is connected to an AC source of emf 220 V and frequency 50 Hz. The instantaneous voltage of the source is 0 V when the peak value of current is ${{\sqrt a } \over \pi }$ A. The value of $a$ is ___________.
Explanation:
${I_{rms}} = {{{V_{rms}}} \over z}$
$z = {X_2} = {\omega _2}$
$ = 2\pi \times 50 \times {{200} \over {1000}}$
$ = 20\,\pi $
$\therefore$ ${I_{rms}} = {{220} \over {20\pi }} = {{11} \over \pi }$
$\therefore$ ${I_{peak}} = \sqrt 2 \times {{11} \over \pi }$
$ = {{\sqrt {2 \times 121} } \over \pi }$
$ = {{\sqrt {242} } \over \pi }$
A circuit element $\mathrm{X}$ when connected to an a.c. supply of peak voltage $100 \mathrm{~V}$ gives a peak current of $5 \mathrm{~A}$ which is in phase with the voltage. A second element $\mathrm{Y}$ when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by $\frac{\pi}{2}$. If $\mathrm{X}$ and $\mathrm{Y}$ are connected in series to the same supply, what will be the rms value of the current in ampere?
An alternating emf $\mathrm{E}=440 \sin 100 \pi \mathrm{t}$ is applied to a circuit containing an inductance of $\frac{\sqrt{2}}{\pi} \mathrm{H}$. If an a.c. ammeter is connected in the circuit, its reading will be :
A coil of inductance 1 H and resistance $100 \,\Omega$ is connected to a battery of 6 V. Determine approximately :
(a) The time elapsed before the current acquires half of its steady - state value.
(b) The energy stored in the magnetic field associated with the coil at an instant 15 ms after the circuit is switched on. (Given $\ln 2=0.693, \mathrm{e}^{-3 / 2}=0.25$)
A transformer operating at primary voltage $8 \,\mathrm{kV}$ and secondary voltage $160 \mathrm{~V}$ serves a load of $80 \mathrm{~kW}$. Assuming the transformer to be ideal with purely resistive load and working on unity power factor, the loads in the primary and secondary circuit would be
The equation of current in a purely inductive circuit is $5 \sin \left(49\, \pi t-30^{\circ}\right)$. If the inductance is $30 \,\mathrm{mH}$ then the equation for the voltage across the inductor, will be :
$\left\{\right.$ Let $\left.\pi=\frac{22}{7}\right\}$
A series LCR circuit has $\mathrm{L}=0.01\, \mathrm{H}, \mathrm{R}=10\, \Omega$ and $\mathrm{C}=1 \mu \mathrm{F}$ and it is connected to ac voltage of amplitude $\left(\mathrm{V}_{\mathrm{m}}\right) 50 \mathrm{~V}$. At frequency $60 \%$ lower than resonant frequency, the amplitude of current will be approximately :
A direct current of $4 \mathrm{~A}$ and an alternating current of peak value $4 \mathrm{~A}$ flow through resistance of $3\, \Omega$ and $2\,\Omega$ respectively. The ratio of heat produced in the two resistances in same interval of time will be :
In a series $L R$ 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:
When you walk through a metal detector carrying a metal object in your pocket, it raises an alarm. This phenomenon works on :
To increase the resonant frequency in series LCR circuit,
In series RLC resonator, if the self inductance and capacitance become double, the new resonant frequency (f2) and new quality factor (Q2) will be :
(f1 = original resonant frequency, Q1 = original quality factor)
For a series LCR circuit, I vs $\omega$ curve is shown :
(a) To the left of $\omega$r, the circuit is mainly capacitive.
(b) To the left of $\omega$r, the circuit is mainly inductive.
(c) At $\omega$r, impedance of the circuit is equal to the resistance of the circuit.
(d) At $\omega$r, impedance of the circuit is 0.
Choose the most appropriate answer from the options given below :
If L, C and R are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?
The current flowing through an ac circuit is given by
I = 5 sin(120$\pi$t)A
How long will the current take to reach the peak value starting from zero?
A sinusoidal voltage V(t) = 210 sin 3000 t volt is applied to a series LCR circuit in which L = 10 mH, C = 25 $\mu$F and R = 100 $\Omega$. The phase difference ($\Phi $) between the applied voltage and resultant current will be :
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 :