Electromagnetic Induction and Alternating Currents — JEE Main PYQs

The figure given below shows an $LCR$ series circuit with two switches $S_1$ and $S_2$. When switch $S_1$ is closed keeping $S_2$ open, the phase difference ($\phi$) between the current and source voltage is $30°$ and phase difference is $60°$ when $S_2$ is closed keeping $S_1$ open. The value of $(3L_1 - L_2)$ is ________ H.

[Figure: Series circuit with capacitor $C = 100\ \mu F$ and resistor $R$; switch $S_1$ in series with inductor $L_2$ on one branch, switch $S_2$ in series with inductor $L_1$ on another branch; AC source $v = V_0 \sin(300t)$.]

A circular current loop of radius $R$ is placed inside square loop of side length $L$ ($L \gg R$) such that they are co-planar and their centers coincide. The permeability of free space is $\mu_0$. The mutual inductance between circular loop and square loop is ________.

An inductor of 10 mH, capacitor of 0.1 μF and a resistor of 100 Ω are connected in series across an a.c power supply 220 V, 70 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 a.c. source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current is observed as $I$. Now the frequency of the source is changed to $\omega/4$, (keeping the voltage unchanged) the current is found to be $I/3$. The ratio of resistance to reactance at frequency $\omega$ is

In the given circuit below inductance values of $L_1$, $L_2$ and $L_3$ are same. The magnetic energy stored in the entire circuit is ($U_t$) and that stored in the $L_2$ inductor is ($U_l$). $U_t / U_l$ is ________.

(Ignore the mutual inductance if any)

[Figure: Inductor $L_1$ in series with a parallel combination of inductors $L_2$ (upper branch) and $L_3$ (lower branch).]

A metal rod of length $L$ rotates about one end at origin with a uniform angular velocity $\omega$. The magnetic field radially falls off as $B(r) = B_0\, e^{-\lambda r}$; $\lambda$ being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is :

A series LCR circuit with $R = 20\ \Omega$, $L = 1.6 \text{ H}$ and $C = 40\ \mu\text{F}$ is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is _________ $\Omega$.

A circular loop of radius $20 \text{ cm}$ and resistance $2\ \Omega$ is placed in a time varying magnetic field $\vec{B} = \left( 2t^2 + 2t + 3 \right) \text{ T}$. At $t = 0$, for the plane of the loop being perpendicular to the magnetic field and, the induced current in the loop at $t = 3 \text{ s}$ is $\dfrac{\alpha}{50} \text{ A}$. The value of $\alpha$ is __________.

(Take $\pi = 22/7$)

An inductor of inductance 10 mH having resistance of 100 $\Omega$ is connected to battery of E.M.F. 1.0 V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is 2 mA and 4 mA is ________.

[Figure: A series circuit with an inductor, a battery, and an open switch]

A LCR series circuit driven with $E_{rms} = 90$ V at frequency $f_d = 30$ 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 ______.

A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as $B = 0.4\sin(300t)$ Tesla. The normal to the plane of loop makes an angle of $60^\circ$ with the field. The maximum induced emf produced in the loop is __________ mV.

A $30\,\text{cm}$ long solenoid has 10 turns per cm and area of $5\,\text{cm}^2$. The current through the solenoid coil varies from $2\,\text{A}$ to $4\,\text{A}$ in $3.14\,\text{s}$. The e.m.f. induced in the coil is $\alpha \times 10^{-5}\,\text{V}$. The value $\alpha$ is ________.

A solenoid having area A and length '$l$' is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is :

An electric bulb rated as 100W-220V is connected to an ac source of rms voltage 220V. The peak value of current through the bulb is :

An inductor of self inductance 1 H is connected in series with a resistor of $100\,\pi$ ohm and an ac supply of $100\,\pi$ volt, 50 Hz. Maximum current flowing in the circuit is ________ A.

An ac current is represented as

$$i = 5\sqrt{2} + 10\cos\left(650\pi t + \frac{\pi}{6}\right)\ \mathrm{Amp}$$

The r.m.s value of the current is

For ac circuit shown in figure, $\mathrm{R} = 100\ \mathrm{k\Omega}$ and $\mathrm{C} = 100\ \mathrm{pF}$ and the phase difference between $\mathrm{V_{in}}$ and $(\mathrm{V_B} - \mathrm{V_A})$ is $90°$. The input signal frequency is $10^x$ rad/sec, where '$x$' is ______.

[Figure: an ac source $V_{in}$ driving two parallel branches — one branch has R in series with C to ground with node $V_A$ between them, the other has C in series with R to ground with node $V_B$ between them]

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 __________ W.

A rectangular metallic loop is moving out of a region of uniform magnetic field into a magnetic field-free region with a constant velocity. When the loop is partially inside the magnetic field, the plot of induced emf ($\varepsilon$) with time (t) is given by :

A series LCR circuit is connected to an alternating source of emf E. The current amplitude at resonant frequency is $I_0$. If the value of resistance R becomes twice of its initial value then amplitude of current at resonance will be

Regarding self-inductance:

A. The self-inductance of the coil depends on its geometry.

B. Self-inductance does not depend on the permeability of the medium.

C. Self-induced e.m.f. opposes any change in the current in a circuit.

D. Self-inductance is electromagnetic analogue of mass in mechanics.

E. Work needs to be done against self-induced e.m.f. in establishing the current.

Choose the correct answer from the options given below:

In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 $\Omega$, the value of the current in the circuit will be _____ A.

[Figure: series circuit with a 12 V battery, a 3 H inductor, and a rheostat R (resistor with sliding contact)]

In a series LCR circuit, a resistor of $300~\Omega$, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is __________ $\times 10^4$ radians $\mathrm{s^{-1}}$.

An alternating current is given by $I = I_A \sin\omega t + I_B \cos\omega t$. The r.m.s current will be

A uniform magnetic field of 0.4 T acts perpendicular to a circular copper disc 20 cm in radius. The disc is having a uniform angular velocity of $10\pi\,\mathrm{rad\,s^{-1}}$ about an axis through its centre and perpendicular to the disc. What is the potential difference developed between the axis of the disc and the rim? ($\pi = 3.14$)

[Figure: a conducting bar moving on two conducting rails that form a V-shape (vertex at bottom); a constant magnetic field B is directed into the page (shown by × symbols)]

A conducting bar moves on two conducting rails as shown in the figure. A constant magnetic field B exists into the page. The bar starts to move from the vertex at time $t = 0$ with a constant velocity. If the induced EMF is $E \propto t^n$, then value of n is __________.

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to :

A. pure inductor.

B. pure capacitor.

C. pure resistor.

D. combination of an inductor and capacitor.

Choose the correct answer from the options given below :

A alternating current at any instant is given by $i = \left[6 + \sqrt{56}\,\sin\left(100\pi \mathrm{t} + \pi/3\right)\right]$ A. The rms value of the current is ________ A.

Match List I with List II

	LIST I		LIST II
A.	Purely capacitive circuit	I.	[Phasor diagram: current $I$ along the vertical axis, voltage $V$ along the horizontal axis, with a $90°$ angle between them ($I$ leads $V$ by $90°$)]
B.	Purely inductive circuit	II.	[Phasor diagram: $V$ and $I$ both along the same horizontal direction (in phase)]
C.	LCR series at resonance	III.	[Phasor diagram: $V$ at an angle $\theta$ above the horizontal $I$ axis ($V$ leads $I$ by $\theta$)]
D.	LCR series circuit	IV.	[Phasor diagram: voltage $V$ along the vertical axis, current $I$ along the horizontal axis, with a $90°$ angle between them ($V$ leads $I$ by $90°$)]

Choose the correct answer from the options given below:

A rod of length 60 $cm$ rotates with a uniform angular velocity 20 rad $s^{-1}$ about its perpendicular bisector, in a uniform magnetic filed $0.5T$. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is ______ V.

Two conducting circular loops A and B are placed in the same plane with their centres coinciding as shown in figure. The mutual inductance between them is :

[Figure: two concentric coplanar circular loops — small inner loop A of radius a and large outer loop B of radius b, with b >> a]

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $20\ \mu\mathrm{F}$ is ________ V.

[Figure: series LCR circuit with L = 1H, C = 20 μF, R = 300 Ω connected to an ac source $V = 50\sqrt{2} \sin 100t$ volt]

A series LCR circuit is subjected to an ac signal of 200 V, 50 Hz. If the voltage across the inductor ($L = 10\ \mathrm{mH}$) is 31.4 V, then the current in this circuit is ________.

The current in an inductor is given by $I = (3t + 8)$ where t is in second. The magnitude of induced emf produced in the inductor is 12 mV. The self-inductance of the inductor ________ mH.

Given below are two statements :

Statement I : In an LCR series circuit, current is maximum at resonance.

Statement II : Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltage source.

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

When a $dc$ voltage of 100V is applied to an inductor, a $dc$ current of 5A flows through it. When an $ac$ voltage of 200V peak value is connected to inductor, its inductive reactance is found to be $20\sqrt{3}\,\Omega$. The power dissipated in the circuit is _________ $W$.

In a coil, the current changes from $-2\ \mathrm{A}$ to $+2\ \mathrm{A}$ in 0.2 s and induces an emf of 0.1 V. The self inductance of the coil is :

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is 2.5 nF. If resistance of 200Ω and 100 mH inductor is being used in the given circuit. The frequency of ac source is ________ $\times 10^3$ Hz. (given $\pi^2 = 10$)

A LCR circuit is at resonance for a capacitor C, inductance L and resistance R. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

A square loop PQRS having 10 turns, area $3.6 \times 10^{-3}\ \mathrm{m^2}$ and resistance $100\ \Omega$ is slowly and uniformly being pulled out of a uniform magnetic field of magnitude B = 0.5 T as shown. Work done in pulling the loop out of the field in 1.0 s is ________ $\times 10^{-6}$ J.

[Figure: square loop PQRS inside a region of uniform magnetic field directed into the page (shown by crosses), being pulled out to the left with velocity $v$]

A coil of negligible resistance is connected in series with $90\ \Omega$ resistor across $120\ \mathrm{V}$, $60\ \mathrm{Hz}$ supply. A voltmeter reads $36\ \mathrm{V}$ across resistance. Inductance of the coil is :

An alternating emf $\mathrm{E} = 110\sqrt{2}\ \sin 100\mathrm{t}$ volt is applied to a capacitor of $2\ \mu\mathrm{F}$, the rms value of current in the circuit is __________ mA.

A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb :

When a coil is connected across a 20 V dc supply, it draws a current of 5 A. When it is connected across 20 V, 50 Hz ac supply, it draws a current of 4 A. The self inductance of the coil is ________ mH. (Take $\pi = 3$)

A square loop of side 15 cm being moved towards right at a constant speed of 2 cm/s as shown in figure. The front edge enters the 50 cm wide magnetic field at $t = 0$. The value of induced emf in the loop at $t = 10$ s will be :

[Figure: a 15 cm square loop moving right at 2 cm/s towards a 50 cm wide region of uniform magnetic field $B = 1.0~\mathrm{T}$ directed into the page]

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 $8\sqrt{2}$ V. The power dissipation in the circuit is ________ W.

In series LCR circuit, the capacitance is changed from C to 4C. To keep the resonance frequency unchanged, the new inductance should be :

A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the $x$-axis and $y$-axis respectively, moves with a velocity of 5 cm/s in the positive $x$ axis direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10^{-3}$ T/cm along the negative $x$ direction and it is decreasing with time at the rate of $10^{-3}$ T/s. If the resistance of the loop is 6 m$\Omega$, the power dissipated by the loop as heat is ________ $\times 10^{-9}$ W.

A transformer has an efficiency of 80% and works at 10 V and 4 kW. If the secondary voltage is 240 V, then the current in the secondary coil is :

A coil of 200 turns and area 0.20 $\mathrm{m}^2$ is rotated at half a revolution per second and is placed in uniform magnetic field of 0.01 T perpendicular to axis of rotation of the coil. The maximum voltage generated in the coil is $\frac{2\pi}{\beta}$ volt. The value of $\beta$ is ________.

A rectangular loop of length 2.5 m and width 2 m is placed at $60^{\circ}$ to a magnetic field of 4 T. The loop is removed from the field in 10 s. The average emf induced in the loop during this time is :

Two coils have mutual inductance 0.002 H. The current changes in the first coil according to the relation $i = i_0 \sin \omega t$, where $i_0 = 5$A and $\omega = 50\pi$ rad/s. The maximum value of emf in the second coil is $\frac{\pi}{\alpha}$ V. The value of $\alpha$ is ________.

A capacitor of capacitance 100 $\mu$F is charged to a potential of 12 V and connected to a 6.4 mH inductor to produce oscillations. The maximum current in the circuit would be :

A square loop of side 10 cm and resistance $0.7\,\Omega$ is placed vertically in east-west plane. A uniform magnetic field of 0.20 $T$ is set up across the plane in north east direction. The magnetic field is decreased to zero in 1 s at a steady rate. Then, magnitude of induced emf is $\sqrt{x} \times 10^{-3}\,V$. The value of $x$ is __________.

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

Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistances shown in figure. The output voltage $(\mathrm{V}_0)$ is :

[Figure: 220 V ac source on transformer primary; secondary connected to a $15~\mathrm{k}\Omega$ resistor in series with a $7~\mathrm{k}\Omega$ resistor; $\mathrm{V}_0$ taken across the $7~\mathrm{k}\Omega$ resistor]

A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and angle of dip is $30°$. The emf induced across the blades is $\mathrm{N}\pi \times 10^{-5}~\mathrm{V}$. The value of N is ________.

An alternating voltage $V(t) = 220 \sin 100\pi t$ volt is applied to a purely resistive load of $50\ \Omega$. The time taken for the current to rise from half of the peak value to the peak value is:

A power transmission line feeds input power at $2.3\ kV$ to a step down transformer with its primary winding having $3000$ turns. The output power is delivered at $230\ V$ by the transformer. The current in the primary of the transformer is $5A$ and its efficiency is $90\%$. The winding of transformer is made of copper. The output current of transformer is _______ $A$.

A coil is places perpendicular to a magnetic field of 5000 T. When the field is changed to 3000 T in 2 s, an induced emf of 22 V is produced in the coil. If the diameter of the coil is 0.02 m, then the number of turns in the coil is:

A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L$ $(L = l^2)$. The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{-7}\ H$, where $x =$ ______.

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:

The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8\ \Omega$ varies with time (in seconds) as $\phi = 5t^2 - 36t + 1$. The induced current in the circuit at $t = 2$ s is ________ $A$.