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JEE Mains | PYQ

A coil of area $A$ and $N$ turns is rotating with angular velocity $\omega$ in a uniform magnetic field $\vec{B}$ about an axis perpendicular to $\vec{B}$. Magnetic flux $\phi$ and induced emf $\varepsilon$ across it, at an instant when $\vec{B}$ is parallel to the plane of coil, are:

Magnetic flux (in weber) in a closed circuit of resistance \( 20 \, \Omega \) varies with time \( t \, (\text{s}) \) as \( \phi = 8t^{2} - 9t + 5 \). The magnitude of the induced current at \( t = 0.25 \, \text{s} \) will be ________ \( \text{mA} \).

An oscillating LC circuit consists of a \( 75 \, \text{mH} \) inductor and a \( 1.2 \, \mu \text{F} \) capacitor. If the maximum charge to the capacitor is \( 2.7 \, \mu \text{C} \). The maximum current in the circuit will be ______ \( \text{mA} \).

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:

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 $\sqrt{x} \times 10^{-7}\text{ H}$, where $x = \text{________}$.

For the given circuit the current through battery of \( 6 \, \text{V} \) just after closing the switch ‘\( S \)’ will be _____ \( \text{A} \).
 

The magnetic field in a region is given by \( \vec{B} = B_{0} \left( \dfrac{x}{a} \right) \hat{k} \). A square loop of side \( d \) is placed with its edges along the \( x \) and \( y \) axes. The loop is moved with a constant velocity \( \vec{v} = v_{0} \hat{i} \). The emf induced in the loop is:

A coil having \( 100 \) turns, area of \( 5 \times 10^{-3} \, \text{m}^2 \), carrying current of \( 1 \, \text{mA} \) is placed in uniform magnetic field of \( 0.20 \, \text{T} \) such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through \( 90^\circ \) is ________ \( \mu \text{J} \).

A wire of length \( 1\,\text{m} \) moving with velocity \( 8\,\text{m/s} \) at right angles to a magnetic field of \( 2\,\text{T} \). The magnitude of induced emf, between the ends of wire will be ______.

Figure shows a circuit that contains four identical resistors with resistance \( R = 2.0 \, \Omega \), two identical inductors with inductance \( L = 20 \, \text{mH} \) and an ideal battery with emf \( E = 9 \, \text{V} \). The current ‘\( i \)’ just after the switch ‘\( S \)’ is closed will be:

A capacitor \( C \) is first charged fully with potential difference of \( V_0 \) and disconnected from the battery. The charged capacitor is connected across an inductor having inductance \( L \). In \( t \) \( \text{s} \) \( 25\% \) of the initial energy in the capacitor is transferred to the inductor. The value of \( t \) is ______ \( \text{s} \).

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

A massless square loop, of wire of resistance \( 10 \, \Omega \), surrounding a mass of \( 1 \, \text{g} \), hangs vertically with one of its sides in a uniform magnetic field of \( 10^{3} \, \text{G} \), directed outwards in the shaded region. A dc voltage \( V \) is applied to the loop. For what value of \( V \), the magnetic force will exactly balance the weight of the supporting mass of \( 1 \, \text{g} \)? \( \text{(If sides of the loop = } 10 \, \text{cm}, \, g = 10 \, \text{ms}^{-2} \text{)} \)

A constant magnetic field of \( 1\,\text{T} \) is applied in the \( x > 0 \) region. A metallic circular ring of radius \( 1\,\text{m} \) is moving with a constant velocity of \( 1\,\text{m/s} \) along the \( x \)-axis. At \( t = 0\,\text{s} \), the centre \( O \) of the ring is at \( x = -1\,\text{m} \). What will be the value of the induced emf in the ring at \( t = 1\,\text{s} \)? (Assume the velocity of the ring does not change)

In the given figure, an inductor and a resistor are connected in series with a battery of emf \( E \) volt. \( \dfrac{E^{a}}{2b} \, \text{J/s} \) represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of \( \dfrac{b}{a} \) will be ______.

The induced emf can be produced in a coil by:
A. moving the coil with uniform speed inside uniform magnetic field
B. moving the coil with non-uniform speed inside uniform magnetic field
C. rotating the coil inside the uniform magnetic field
D. changing the area of the coil inside the uniform magnetic field
Choose the correct answer from the options given below:

A coil of inductance \( 1 \, \text{H} \) and resistance \( 100\Omega \) is connected to a battery of \( 6 \, \text{V} \). 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 \( 15 \, \text{ms} \) after the circuit is switched on (Given \( \ln 2 = 0.693 \), \( e^{-3/2} = 0.25 \))

A square loop of side \( 20 \, \text{cm} \) and resistance \( 1 \, \Omega \) is moved towards right with a constant speed \( v_0 \). The right arm of the loop is in a uniform magnetic field of \( 5 \, \text{T} \). The field is perpendicular to the plane of the loop and is going into it. The loop is connected to a network of resistors each of value \( 4 \, \Omega \). What should be the value of \( v_0 \) so that a steady current of \( 2 \, \text{mA} \) flows in the loop?

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

When a coil is placed in a time dependent magnetic field the power dissipated in it is \( P \). The number of turns, area of the coil and radius of the coil wire are \( N \), \( A \) and \( r \) respectively. For a second coils number of turns, area of the coil and radius of the coil wire are \( 2N \), \( 2A \) and \( 3r \) respectively. When the first coil is replaced with second coil the power dissipated in it is \( \sqrt{2}\, \alpha P \). The value of \( \alpha \) is ______.

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 ________.

Consider \( I_1 \) and \( I_2 \) are the currents flowing simultaneously in two nearby coils \( 1 \) and \( 2 \), respectively. If \( L_1 \) = self inductance of coil \( 1 \), \( M_{12} \) = mutual inductance of coil \( 1 \) with respect to coil \( 2 \), then the value of induced emf in coil \( 1 \) will be:

Three identical resistors with resistance \( R = 12 \, \Omega \) and two identical inductors with self inductance \( L = 5 \, \text{mH} \) are connected to an ideal battery with emf of \( 12 \, \text{V} \) as shown in figure. The current through the battery long after the switch has been closed will be ________ \( \text{A} \).

A \( 10 \, \Omega \), \( 20 \,\mathrm{mH} \) coil carrying constant current is connected to a battery of \( 20 \, \mathrm{V} \) through a switch. Now after switch is opened current becomes zero in \( 100 \, \mu \mathrm{s} \). The average e.m.f. induced in the coil is _____ \( \mathrm{V} \).

In an ac generator, a rectangular coil of \( 100 \) turns each having area \( 14 \times 10^{-2}\, \text{m}^2 \) is rotated at \( 360 \, \text{rev/min} \) about an axis perpendicular to a uniform magnetic field of magnitude \( 3.0 \, \text{T} \). The maximum value of the emf produced will be ______ \( \text{V} \). (Take \( \pi = \frac{22}{7} \))

Figure shows the circuit that contains three resistances \(9 \, \Omega\) each and two inductors \((4 \, \text{mH}\) each). The reading of ammeter at the moment switch \( K \) is turned ON, is ________ \( \text{A} \).

Conductor wire \( ABCDE \) with each arm \( 10 \, \text{cm} \) in length is placed in magnetic field of \( \dfrac{1}{\sqrt{2}} \, \text{Tesla} \), perpendicular to its plane. When conductor is pulled towards right with constant velocity of \( 10 \, \text{cm/s} \), induced emf between points \( A \) and \( E \) is _______ \( \text{mV} \).

A coil has an inductance of \( 2 \, \text{H} \) and resistance of \( 4 \, \Omega \). A \( 10 \, \text{V} \) is applied across the coil. The energy stored in the magnetic field after the current has built up to its equilibrium value will be ________ \( \times 10^{-2} \, \text{J} \).

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 __________.

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

For the given circuit the current \( i \) through the battery when the key in closed and the steady state has been reached is:

A tightly wound long solenoid carries a current of \( 1.5 \, \text{A} \). An electron is executing uniform circular motion inside the solenoid with a time period of \( 75 \, \text{ns} \). The number of turns per metre in the solenoid is ________.
[Take mass of electron \( m_e = 9 \times 10^{-31} \, \text{kg} \), charge of electron \( |q_e| = 1.6 \times 10^{-19} \, \text{C} \), \( \mu_0 = 4\pi \times 10^{-7} \, \frac{N}{A^{2}} \), \( 1 \, \text{ns} = 10^{-9} \, \text{s} \)]

For the given circuit, comment on the type of transformer used.

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 \((U_t)\). \( \dfrac{U_{t}}{U_{l}} \) is ______.
(Ignore the mutual inductance if any)

The arm PQ of a rectangular conductor is moving from \(x = 0\) to \(x = 2b\) outwards and then inwards from \(x = 2b\) to \(x = 0\) as shown in the figure. A uniform magnetic field perpendicular to the plane is acting from \(x = 0\) to \(x = b\). Identify the graph showing the variation of different quantities with distance.

A circular coil of \( 1000 \) turns each with area \( 1 \, \text{m}^{2} \) is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of \( 0.07 \, \text{T} \). The maximum voltage generation will be ________ \( \text{V} \).

Two concentric circular with radii \( 1 \, \text{cm} \) and \( 1000 \, \text{cm} \), and number of turns \( 10 \) and \( 200 \) respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be __________ \( \times 10^{-8} \, \text{H} \). (Take, \( \pi^{2} = 10 \))

A \( 12 \, \text{V} \) battery connected to a coil of resistance \( 6 \, \Omega \) through a switch, drives a constant current in the circuit. The switch is opened in \( 1 \, \text{ms} \). The emf induced across the coil is \( 20 \, \text{V} \). The inductance of the coil is:

A bar magnet is passing through a conducting loop of radius \( R \) with velocity \( v \). The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve:

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

A coil is placed in a magnetic field \( \vec{B} \) as shown below:

A current is induced in the coil because \( \vec{B} \) is:
A current is induced in the coil because \( \vec{B} \) is:

An insulated copper wire of \( 100 \) turns is wrapped around a wooden cylindrical core of the cross-sectional area \( 24 \, \text{cm}^2 \). The two ends of the wire are connected to a resistor. The total resistance in the circuit is \( 12 \, \Omega \). If an externally applied uniform magnetic field in the core along its axis changes from \( 1.5 \, \text{T} \) in one direction to \( 1.5 \, \text{T} \) in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be ____________ \( \text{mC} \).

A metallic cube of side \( 15 \, \text{cm} \) moving along \( y \)-axis at a uniform velocity of \( 2 \, \text{m s}^{-1} \). In a region of uniform magnetic field of magnitude \( 0.5 \, \text{T} \) directed along \( z \)-axis. In equilibrium the potential difference between the faces of higher and lower potential developed because of the motion through the field will be ________ \( \text{mV} .\)

A conducting circular loop of radius \( \frac{10}{\sqrt{\pi}} \, \text{cm} \) is placed perpendicular to a uniform magnetic field of \( 0.5 \, \text{T} \). The magnetic field is decreased to zero in \( 0.5 \, \text{s} \) at a steady rate. The induced emf in the circular loop at \( 0.05 \, \text{s} \) is:

A \( 20 \, \text{cm} \) long metallic rod is rotated with \( 210 \, \text{rpm} \) about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field \( 0.2 \, \text{T} \) parallel to the axis exists everywhere. The emf developed between the centre and the ring is __________ \( \text{mV} \).
Take \( \pi = \dfrac{22}{7} \)

Match List I with List – II

Choose the CORRECT answer from the options given below:

A uniform magnetic field of \( 0.4 \, \mathrm{T} \) acts perpendicular to a circular copper disc \( 20 \, \mathrm{cm} \) 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? \( (\pi = 3.14) \)

XPQY is a vertical smooth long loop having a total resistance \( R \) where \( PX \) is parallel to \( QY \) and separation between them is \( l \). A constant magnetic field \( B \) perpendicular to the plane of the loop exists in the entire space. A rod \( CD \) of length \( L \) (\( L > l \)) and mass \( m \) is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is _______ \( \text{m/s} \). (\( g = \) acceleration due to gravity)

A conducting circular loop is placed in a uniform magnetic field of \( 0.4 \, \text{T} \) with its plane perpendicular to the field. Somehow, the radius of the loop starts expanding at a constant rate of \( 1 \, \text{mm/s} \). The magnitude of induced emf in the loop at an instant when the radius of loop is \( 2 \, \text{cm} \) will be ___ \( \mu \text{V} \).

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:

Given below are two statements:
Statement I: A time varying electric field is a source of changing magnetic field and vice-versa. Thus a disturbance in electric or magnetic field creates EM waves.
Statement II: In a material medium, the EM wave travels with speed \( v = \frac{1}{\sqrt{\mu_0 \, \varepsilon_0}} \). In the light of the above statements, choose the correct answer from the options given below.

A transformer operating at primary voltage \( 8 \, \text{kV} \) and secondary voltage \( 160 \, \text{V} \) serves a load of \( 80 \, \text{kW} \). 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:

A \( 1 \, \text{m} \) long metal rod XY completes the circuit as shown in figure. The plane of the circuit is perpendicular to the magnetic field of flux density \( 0.15 \, \text{T} \). If the resistance of the circuit is \( 5 \, \Omega \), the force needed to move the rod in direction, as indicated, with a constant speed of \( 4 \, \text{m/s} \) will be ______ \( 10^{-3} \, \text{N} \).

A circular coil of radius \( 8.0 \, \text{cm} \) \( 20 \) turns is rotated about its vertical diameter with an angular speed of \( 50 \, \text{rad s}^{-1} \) in a uniform horizontal magnetic field of \( 3.0 \times 10^{-2} \, \text{T} \). The maximum emf induced the coil will be __________ \( \times 10^{-2} \) volt (rounded off to the nearest integer).

A metallic conductor of length \( 1 \, \text{m} \) rotates in a vertical plane parallel to east-west direction about one of its end with angular velocity \( 5 \, \text{rad s}^{-1} \). If the horizontal component of earth's magnetic field is \( 0.2 \times 10^{-4} \, \text{T} \), then emf induced between the two ends of the conductor is:

An inductor coil stores \( 64 \, \text{J} \) of magnetic field energy and dissipates energy at the rate of \( 640 \, \text{W} \) when a current of \( 8 \, \text{A} \) is passed through it. If this coil is joined across an ideal battery, find the time constant of the circuit in seconds:

In a coil of resistance \( 8 \, \Omega \), the magnetic flux to an external magnetic field varies with time as \( \phi = \dfrac{2}{3} \left( 9 - t^{2} \right) \). The value of total heat produced in the coil, till the flux becomes zero, will be __________ \( \text{J} \).

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

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 circular conducting coil of radius \( 1 \, \text{m} \) is being heated by the change of magnetic field \( \vec{B} \) passing perpendicular to the plane in which the coil is laid. The resistance of the coil is \( 2 \, \mu \Omega \). The magnetic field is slowly switched off such that its magnitude changes in time as \( B = \dfrac{4}{\pi} \times 10^{-3} \, \text{T} \left( 1 - \dfrac{t}{100} \right) \). The energy dissipated by the coil before the magnetic field is switched off completely is \( E = \) __________ \( \text{m} \)).

The source of time varying magnetic field may be:
(A) a permanent magnet   (B) an electric field changing linearly with time 
 (C) direct current                (D) a decelerating charge particle 
 (E) an antenna fed with a digital signal
Choose the correct answer from the options given below:

The energy of an electromagnetic wave contained in a small volume oscillates with:

A square loop of side \( 2.0 \, \text{cm} \) is placed inside a long solenoid that has \( 50 \) turns per centimeter and carries a sinusoidally varying current of amplitude \( 2.5 \, \mathring{A}. \) and angular frequency \( 700 \, \text{rad s}^{-1} \). The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is \( x \times 10^{-4} \, \text{V} \). The value of \( x \) is ______.

A \( 30 \ \text{cm} \) long solenoid has \( 10 \) turns per \( \text{cm} \) 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 conducting circular loop is placed in X–Y plane in presence of magnetic field \( \vec{B} = (3 t^3 \, \hat{j} + 3 t^2 \, \hat{k}) \) in SI unit. If the radius of the loop is \( 1 \, \text{m} \), the induced emf in the loop, at time \( t = 2 \, \text{s} \) is \( n\pi \, \text{V} \). The value of \( n \) is _____.

The magnetic flux through a coil perpendicular to its plane is varying according to the relation \( \phi = (5t^3 + 4t^2 + 2t - 5) \) Weber. If the resistance of the coil is \( 5 \, \text{ohm} \), then the induced current through the coil at \( t = 2 \, \text{s} \) will be:

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:

A circular loop of radius \( 20 \, \text{cm} \) and resistance \( 2 \, \Omega \) is placed in a time varying magnetic field \( \vec{B} = (2t^{2} + 2t + 3)\,\text{T} \). 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 \( \pi = \dfrac{22}{7} \))

The equivalent resistance between A and B is:

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:

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

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

Given below are two statements:
Statement I: The diamagnetic property depends on temperature.
Statement II: The induced magnetic dipole moment in a diamagnetic sample is always opposite to the magnetizing field.
In the light of given statements, choose the correct answer from the options given below.

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

For a series LCR circuit with \( R = 100\,\Omega \), \( L = 0.5\,\text{mH} \) and \( C = 0.1\,\text{pF} \) connected across \( 220\,\text{V} - 50\,\text{Hz} \) AC supply, the phase angle between current and supplied voltage and the nature of the circuit is:

The displacement current of \( 4.425 \, \mu\text{A} \) is developed in the space between the plates of parallel plate capacitor when voltage is changing at a rate of \( 10^{6} \, \text{V s}^{-1} \). The area of each plate of the capacitor is \( 40 \, \text{cm}^{2} \). The distance between each plate of the capacitor is \( x \times 10^{-3} \, \text{m} \). The value of \( x \) is, (Permittivity of free space, \( E_{0} = 8.85 \times 10^{-12} \, \text{C}^{2}\text{N}^{-1}\text{m}^{-2} \)) ______.

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

A small square loop of wire of side \( l \) is placed inside a large square loop of wire \( L \) \( (L \gg l) \). Both loops are coplanar and their centres coincide at point O as shown in figure. The mutual inductance of the system 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 \( t = 2 \, \text{s} \) is __________ A.

A metallic rod of length ‘L’ is rotated with an angular speed of \( '\omega' \) normal to a uniform magnetic field ‘B’ about an axis passing through one end of rod as shown in figure. The induced emf will be:

Match List I with List II:

Choose the correct answer from the options given below:

Four identical long solenoids A, B, C and D are connected to each other as shown in the figure. If the magnetic field at the center of A is \( 3 \, \text{T} \), the field at the center of C would be:
(Assume that the magnetic field is confined with in the volume of respective solenoid).

Three identical coils \( C_1 \), \( C_2 \) and \( C_3 \) are closely placed such that they share a common axis. \( C_2 \) is exactly midway. \( C_1 \) carries current \( I \) in anti-clockwise direction while \( C_3 \) carries current \( I \) in clockwise direction. An induced current flows through \( C_2 \) will be in clockwise direction when:

The current in a coil of self inductance \( 2.0 \, \text{H} \) is increasing according to \( I = 2 \sin(t^2) \, \text{A} \). The amount of energy spent during the period when current changes from \( 0 \) to \( 2 \, \text{A} \) is __________ \( \text{J} \).

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

\( N \) equally spaced charges each of value \( q \), are placed on a circle of radius \( R \). The circle rotates about its axis with an angular velocity \( \omega \) as shown in the figure. A bigger Amperian loop \( B \) encloses the whole circle where as a smaller Amperian loop \( A \) encloses a small segment. The difference between enclosed currents, \( I_{A} - I_{B} \) for the given Amperian loops is:

Given below are two statements:
Statement (I): When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account.
Statement (II): Ampere's circuital law does not depend on Biot-Savart's law.
In the light of the above statements, choose the correct answer from the options given below:

Consider an electrical circuit containing a two way switch 'S'. Initially S is open and then \( T_1 \) is connected to \( T_2 \). As the current in \( R = 6 \, \Omega \) attains a maximum value of steady state level, \( T_1 \) is disconnected from \( T_2 \) and immediately connected to \( T_3 \). Potential drop across \( r = 3 \, \Omega \) resistor immediately after \( T_1 \) is connected to \( T_3 \) is _______ \( \text{V} \). (Round off to the Nearest Integer)

A \( 20 \, \text{m} \) long uniform copper wire held horizontally is allowed to fall under the gravity \( (g = 10 \, \text{m/s}^{2}) \) through a uniform horizontal magnetic field of \( 0.5 \, \text{Gauss} \) perpendicular to the length of the wire. The induced EMF across the wire when it travels a vertical distance of \( 200 \, \text{m} \) is ______ mV.

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 \( 12 \, \text{mV} \). The self-inductance of the inductor _____ \( \text{mH} \).

A circular loop of radius \( 7 \, \text{cm} \) is placed in uniform magnetic field of \( 0.2 \, \text{T} \) directed perpendicular to plane of loop. The loop is converted into a square loop in \( 0.5 \, \text{s} \). The EMF induced in the loop is _______ \( \text{mV} \).

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

In the given figure the magnetic flux through the loop increases according to the relation \( \phi_{B}(t) = 10 t^{2} + 20 t \), where \( \phi_{B} \) is in milliwebers and \( t \) is in seconds. The magnitude of current through \( R = 2 \, \Omega \) resistor at \( t = 5 \, \text{s} \) is __________ \( \text{mA} \).

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

A conducting circular loop is rotated about its diameter at a constant angular speed of \( 100 \, \text{rad/s} \) in a magnetic field of \( 0.5 \, \text{T} \) perpendicular to the axis of rotation. When the loop is rotated by \( 30^\circ \) from the horizontal position, the induced EMF is \( 15.4 \, \text{mV} \). The radius of the loop is ______ mm. (Take \( \pi = \frac{22}{7} \))

A conducting circular loop of area \( 1.0 \, \text{m}^{2} \) is placed perpendicular to a magnetic field which varies as \( B = \sin(100 \, t) \) Tesla. If the resistance of the loop is \( 100 \, \Omega \), then the average thermal energy dissipated in the loop in one period is ______________ \( \text{J} \).

A simple pendulum made of mass \( 10 \, \text{g} \) and a metallic wire of length \( 10 \, \text{cm} \) is suspended vertically in a uniform magnetic field of \( 2 \, \text{T} \). The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of \( 60^\circ \) with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ______ \( \text{mV} \). [Take \( g = 10 \, \text{m}/\text{s}^{2} \)]

A horizontal straight wire \( 5 \, \text{m} \) long extending from east to west falling freely at right angle to horizontal component of earths magnetic field \( 0.60 \times 10^{-4} \, \text{wB m}^{-2} \). The instantaneous value of emf induced in the wire when its velocity is \( 10 \, \text{m s}^{-1} \) is __________ \( \times 10^{-3} \, \text{V} \).

The magnetic field \( B \) crossing normally a square metallic plate of area \( 4 \, \text{m}^2 \) is changing with time as shown in figure. The magnitude of induced emf in the plate during \( t = 2 \, \text{s} \) to \( t = 4 \, \text{s} \), is _______ \( \text{mV} \).

Spherical insulating ball and spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following (Assume negligible air friction)

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

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

An emf of \( 0.08 \, \text{V} \) is induced in a metal rod of length \( 10 \, \text{cm} \) held normal to a uniform magnetic field of \( 0.4 \, \text{T} \), when moves with a velocity of:

An inductor of inductance \( 10 \, \text{mH} \) having resistance of \( 100\Omega \) is connected to battery of E.M.F. \( 1.0 \, \text{V} \) 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 \( 2 \, \text{mA} \) and \( 4 \, \text{mA} \) is _____.

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field \( B = 0.8 \, \text{T} \). When released the radius of the loop starts shrinking at a constant rate of \( 2 \, \text{cm s}^{-1} \). The induced emf in the loop at an instant when the radius of the loop is \( 10 \, \text{cm} \) will be ______ \( \text{mV} \).

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