14. Initially plane of coil is parallel to the uniform magnetic field $B$. In time $\Delta t$ it makes to perpendicular to the magnetic field, then charge flows in $\Delta t$ depends on this time as: (1999)
The total charge flowing through the circuit is given by $Q = \frac{\Delta \Phi}{R}$.
This expression shows that the induced charge depends only on the net change in magnetic flux and resistance.
It is independent of the time interval $\Delta t$, meaning it is proportional to $(\Delta t)^0$.
15. The total charge, induced in a conducting loop when it is moved in magnetic field depend on (1992)
Induced charge $Q = \int I dt = \int \frac{e}{R} dt = \int \frac{1}{R} \frac{d\Phi}{dt} dt = \frac{\Delta \Phi}{R}$.
Thus, the total induced charge depends directly on the total change in magnetic flux, $\Delta \Phi$.
It is independent of the rate of change of flux or time.
16. A rectangular coil of $20$ turns and area of cross-section $25 sq.cm$ has a resistance of $100 \Omega$. If a magnetic field which is perpendicular to the plane of coil changes at a rate of $1000 tesla per second$, the current in the coil is (1992)
Induced emf $e = N A \frac{dB}{dt} = 20 \times (25 \times 10^{-4}) \times 1000 = 50 V$.
Induced current $I = \frac{e}{R}$.
$I = \frac{50}{100} = 0.5 A$.
17. Faraday’s laws are consequence of conservation of (1991)
Faraday's laws of electromagnetic induction, and specifically Lenz's law which determines the direction of induced emf, are based on the principle of conservation of energy.
The mechanical work done in moving a magnet or coil against the opposing magnetic force is converted into electrical energy.
18. A magnetic field of $2 \times 10^{-2} T$ acts at right angles to a coil of area $100 cm^2$, with $50$ turns. The average e.m.f. induced in the coil is $0.1 V$, when it is removed from the field in $t sec$. The value of $t$ is (1991)
Initial flux $\Phi_i = N B A = 50 \times (2 \times 10^{-2}) \times (100 \times 10^{-4}) = 10^{-2} Wb$.
Final flux $\Phi_f = 0$. Magnitude of average emf $e = \frac{\Delta \Phi}{t} = \frac{10^{-2}}{t}$.
Given $e = 0.1 V$, we have $0.1 = \frac{10^{-2}}{t}$, which gives $t = \frac{10^{-2}}{0.1} = 0.1 s$.
29. In a region of magnetic induction $B = 10^{-2} tesla$, a circular coil of radius $30 cm$ and resistance $\pi^2 ohm$ is rotated about an axis which is perpendicular to the direction of B and which forms a diameter of the coil. If the coil rotates at $200 rpm$ the amplitude of the alternating current induced in the coil is (1988)
The maximum induced emf is $e_0 = B A \omega = B (\pi r^2) (2\pi f)$.
Here $B = 10^{-2} T$, $r = 0.3 m$, $f = \frac{200}{60} = \frac{10}{3} s^{-1}$, and $R = \pi^2 \Omega$.
Amplitude of induced current is $I_0 = \frac{e_0}{R} = \frac{10^{-2} \times \pi(0.3)^2 \times 2\pi \times \frac{10}{3}}{\pi^2} = 10^{-2} \times 0.09 \times \frac{20}{3} = 6 \times 10^{-3} A = 6 mA$.
30. In which of the following devices, the eddy current effect is not used? (2019)
Eddy currents are utilized in induction furnaces (for melting metals), magnetic braking in trains, and dead-beat galvanometers.
An electric heater works purely on the principle of Joule heating due to current flowing through a high-resistance wire element.
Therefore, the eddy current effect is not used in an electric heater.
Eddy currents are circulating currents induced in bulk pieces of conductors (metals).
According to Faraday's law of induction, these currents are set up when there is a change in the magnetic flux linked with the metal.
Hence, they are produced when a bulk metal is placed in a time-varying magnetic field.
20. A wire loop is rotated in a magnetic field. The frequency of change of direction of the induced e.m.f. is: (2013)
The induced emf in a rotating loop varies sinusoidally as $e = e_0 \sin(\omega t)$.
In one complete revolution ($2\pi$ radians), the sine function alternates between positive and negative half-cycles.
Thus, the direction of the induced e.m.f. reverses twice per revolution.
21. A metal ring is held horizontally and bar magnet is dropped through the ring with its length along the axis of the ring. The acceleration of the falling magnet is (1996)
As the magnet falls, magnetic flux linked with the ring increases.
By Lenz's law, an induced current is set up in the ring which opposes the downward motion of the magnet.
An upward repulsive magnetic force acts on the magnet, resulting in an acceleration $a < g$.