A highly conducting ring of radius R is perpendicular to and concentric with the axis of a long solenoid as shown in fig. The ring has a narrow gap of width d in its circumference. The solenoid has a cross-sectional area A and a uniform internal field of magnitude B₀. Now beginning at t = 0, the solenoid current is steadily increased so that the field magnitude at any time t is given by B(t) = B₀ + αt where α > 0. Assuming that no charge can flow across the gap, the end of the ring which has an excess of positive charge and the magnitude of induced e.m.f. in the ring are respectively

(1) X, Aα

(2) X, πR²α

(3) Y, πA²α

(4) Y, πR²α

A wire in the form of a circular loop of radius 10 cm lies in a plane normal to a magnetic field of 100 T. If this wire is pulled to take a square shape in the same plane in 0.1 s, average induced emf in the loop is: 

1.  6.70 volt                   

2.  5.80 volt 

3.  6.75 volt                   

4.  5.75 volt

A superconducting loop of radius R has self inductance L. A uniform and constant magnetic field B is applied perpendicular to the plane of the loop. Initially current in this loop is zero. The loop is rotated by 180°. The current in the loop after rotation is equal to –

1. zero                                 

2. $\frac{{\mathrm{B\pi R}}^2}{\mathrm{L}}$

3. $\frac{2{\mathrm{B\pi R}}^2}{\mathrm{L}}$                           

4. $\frac{{\mathrm{B\pi R}}^2}{2\mathrm{L}}$

A small magnet is along the axis of a coil and its distance from the coil is 80 cm. In this position the flux linked with the coil are $4 \times {10}^{–5}$ weber turns. If the coil is displaced 40 cm towards the magnet in 0.08 second, then the induced emf produced in the coil will be -

1.  0.5 mV                       

2.  1 mV 

3.  7 mV                           

4.  3.5 mV

A long solenoid of diameter \(0.1\) m has \(2\times 10^{4}\) turns per meter. At the centre of the solenoid, a coil of \(100\) turns and a radius of \(0.01\) m is placed with its axis coinciding with the solenoid's axis.  The current in the solenoid reduces at a constant rate from \(0\) A to \(4\) A in \(0.05\) s. If the resistance of the coil is \(10\pi^2~\Omega\), the total charge flowing through the coil during this time is:

A metallic ring connected to a rod oscillates freely like a pendulum. If now a magnetic field is applied in the horizontal direction so that the pendulum now swings through the field, the pendulum will

(1) Keep oscillating with the old-time period

(2) Keep oscillating with a smaller time period

(3) Keep oscillating with a larger time period

(4) Come to rest very soon

A conducting square loop of side L and resistance R moves in its plane with a uniform velocity v perpendicular to one of its sides. A magnetic induction B constant in time and space, pointing perpendicular and into the plane of the loop exists everywhere. The current induced in the loop is

(1) $\frac{Blv}{R}$ clockwise

(2) $\frac{Blv}{R}$ anticlockwise

(3) $\frac{2Blv}{R}$ anticlockwise

(4) Zero

A conducting square frame of side \(a\) and a long straight wire carrying current \(i\) are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity \(v\). The emf induced in the frame will be proportional to:
                
1. \(\frac{1}{x^2}\)
2. \(\frac{1}{(2x-a)^2}\)
3. \(\frac{1}{(2x+a)^2}\)
4. \(\frac{1}{(2x-a)\times (2x+a)}\)

A metal rod moves at a constant velocity in a direction perpendicular to its length. A constant, uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity. Select the correct statement (s) from the following : 

1. The entire rod is at the same electric potential 

2. There is an electric field in the rod 

3. The electric potential is highest at the center of the rod and decreases towards its ends 

4. The electric potential is lowest at the center of the rod and increases towards its ends.

A series combination of inductance (L) and resistance (R) is connected to a battery of emf E. The final value of current depends on –

1. L and R                           

2. E and R 

3. E and L                           

4. E, L, and R