Electromagnetic Induction

A conducting wire of parabolic shape, initially $y = x^2$, is moving with velocity $\vec V = V_0\hat i$ in a non-uniform magnetic field $\vec B = B_0\left(1 + \left(\frac yL\right)^\beta\right)\hat k$, as shown in figure. If $V_0$, $B_0$, $L$ and $\beta$ are positive constants and $\Delta\phi$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

A 10 cm long perfectly conducting wire PQ is moving with a velocity 1 cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $L = 1$ mH and a resistance $R = 1\ \Omega$ as shown in figure. The horizontal rails, $L$ and $R$ lie in the same plane with a uniform magnetic field $B = 1$ T perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is $x\times10^{-3}$ A, where the value of $x$ is ______.

[Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: $e^{-1} = 0.37$, where $e$ is base of the natural logarithm]

In the figure below, the switches $S_1$ and $S_2$ are closed simultaneously at $t = 0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current $I$ in the middle wire reaches its maximum magnitude $I_{max}$ at time $t = \tau$. Which of the following statements is (are) true?

A circular insulated copper wire loop is twisted to form two loops of area $A$ and $2A$ as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field $\vec{B}$ points into the plane of the paper. At $t = 0$, the loop starts rotating about the common diameter as axis with a constant angular velocity $\omega$ in the magnetic field. Which of the following options is/are correct?

A source of constant voltage $V$ is connected to a resistance $R$ and two ideal inductors $L_1$ and $L_2$ through a switch $S$ as shown. There is no mutual inductance between the two inductors. The switch $S$ is initially open. At $t = 0$, the switch is closed and current begins to flow. Which of the following options is/are correct?

In a coil of resistance 100 Ω, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is :

A small circular loop of wire of radius a is located at the centre of a much larger circular wire loop of radius b. The two loops are in the same plane. The outer loop of radius b carries an alternating current $I=I_o\cos(\omega t)$. The emf induced in the smaller inner loop is nearly :

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90° vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at a constant rate of $10\ \mathrm{A\,s^{-1}}$. Which of the following statement(s) is(are) true?

Two inductors $L_1$ (inductance 1 mH, internal resistance 3 Ω) and $L_2$ (inductance 2 mH, internal resistance 4 Ω), and a resistor $R$ (resistance 12 Ω) are all connected in parallel across a 5 V battery. The circuit is switched on at time $t=0$. The ratio of the maximum to the minimum current $(I_{\max}/I_{\min})$ drawn from the battery is

A rigid wire loop of square shape having side of length $L$ and resistance $R$ is moving along the $x$-axis with a constant velocity $v_0$ in the plane of the paper. At $t=0$, the right edge of the loop enters a region of length $3L$ where there is a uniform magnetic field $B_0$ into the plane of the paper, as shown in the figure. For sufficiently large $v_0$, the loop eventually crosses the region. Let $x$ be the location of the right edge of the loop. Let $v(x)$, $I(x)$ and $F(x)$ represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of $x$. Counter-clockwise current is taken as positive.

Which of the following schematic plot(s) is(are) correct? (Ignore gravity)

A series LR circuit is connected to a voltage source with $V(t)=V_0\sin\Omega t$. After very large time, current I(t) behaves as $\left(t_0>>\dfrac LR\right)$ :

Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed '$v$' in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities $\sigma_1$ and $\sigma_2$ are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects) :

A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15 m and height 5 m is flying towards east over Delhi. Its speed is 240 ms$^{-1}$. The earth's magnetic field over Delhi is $5\times10^{-5}$ T with the declination angle ~0° and dip of θ such that $\sin\theta=\dfrac23$. If the voltage developed is $V_B$ between the lower and upper side of the plane and $V_W$ between the tips of the wings then $V_B$ and $V_W$ are close to :

A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as $B=B_0e^{-t/\tau}$, where $B_0$ and τ are constants, at time t = 0. If the resistance of the loop is R then the heat generated in the loop after a long time (t → ∞) is :

An inductor (L = 0.03H) and a resistor (R = 0.15 kΩ) are connected in series to a battery of 15V EMF in a circuit shown below. The key $K_1$ has been kept closed for a long time. Then at t = 0, $K_1$ is opened and key $K_2$ is closed simultaneously. At t = 1ms, the current in the circuit will be : ($e^5\cong150$)

When current in a coil changes from 5 A to 2 A in 0.1 s, an average voltage of 50 V is produced. The self - inductance of the coil is :

In the circuits (a) and (b) switches $S_1$ and $S_2$ are closed at t = 0 and are kept closed for a long time. The variation of currents in the two circuits for t ≥ 0 are roughly shown by (figures are schematic and not drawn to scale) :

In the circuit shown here, the point 'C' is kept connected to point 'A' till the current flowing through the circuit becomes constant. Afterward, suddenly, point 'C' is disconnected from point 'A' and connected to point 'B' at time $t=0$. Ratio of the voltage across resistance and the inductor at $t=L/R$ will be equal to :

A coil of circular cross-section having 1000 turns and 4 cm$^2$ face area is placed with its axis parallel to a magnetic field which decreases by $10^{-2}$ Wb m$^{-2}$ in 0.01 s. The e.m.f. induced in the coil is :

A square frame of side 10 cm and a long straight wire carrying current 1 A are in the plane of the paper. Starting from close to the wire, the frame moves towards the right with a constant speed of 10 ms$^{-1}$ (see figure). The e.m.f induced at the time the left arm of the frame is at $x=10$ cm from the wire is :