8. A conducting circular loop is placed in a uniform magnetic field, $B = 0.025 T$ with its plane perpendicular to the loop. The radius of the loop is made to shrink at a constant rate of $1 mm s^{-1}$. The induced emf when the radius is $2 cm$ is: (2010 Pre)
Flux is $\Phi = B \cdot A = B \cdot \pi r^2$. Magnitude of induced emf is $e = |\frac{d\Phi}{dt}| = B \cdot 2\pi r |\frac{dr}{dt}|$.
Given $B = 0.025 T$, $r = 2 cm = 0.02 m$, and $|\frac{dr}{dt}| = 1 mm s^{-1} = 10^{-3} m s^{-1}$.
$e = 0.025 \cdot 2\pi(0.02) \cdot (10^{-3}) = \pi \times 10^{-6} V = \pi \mu V$.
9. A conducting circular loop is placed in a uniform magnetic field $0.04 T$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $2 mm/s$. The induced emf in the loop when the radius is $2 cm$ is: (2009)
10. A circular disc of radius $0.2 meter$ is placed in a uniform magnetic field of induction $\frac{1}{\pi} (\frac{wb}{m^2})$ in such a way that its axis makes an angle of $60^\circ$ with the magnetic field. The magnetic flux linked with the disc is: (2008)
Magnetic flux is given by $\Phi = BA \cos\theta$.
Here, $\theta = 60^\circ$ (angle between the axis, which is the normal to the area, and the magnetic field).
$\Phi = \left(\frac{1}{\pi}\right) \cdot (\pi \cdot 0.2^2) \cdot \cos(60^\circ) = 0.04 \cdot 0.5 = 0.02 Wb$.
12. The magnetic flux through a circuit of resistance $R$ changes by an amount $\Delta \phi$ in a time $\Delta t$. Then the total quantity of electric charge $Q$ that passes any point in the circuit during the time $\Delta t$ is represented by: (2004)
The induced emf is $e = \frac{\Delta \phi}{\Delta t}$.
The induced current is $I = \frac{e}{R} = \frac{\Delta \phi}{R \Delta t}$.
Total charge $Q = I \Delta t = \frac{\Delta \phi}{R \Delta t} \cdot \Delta t = \frac{\Delta \phi}{R}$.
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 $t = \frac{10^{-2}}{0.1} = 0.1 s$.