1. A square loop of side $1 m$ and resistance $1 \Omega$ is placed in a magnetic field of $0.5 T$. If the plane of loop of perpendicular to the direction of a magnetic field, the magnetic flux through the loop is: (2022)
Area of the square loop is $A = 1 \times 1 = 1 m^2$.
Since the plane of the loop is perpendicular to the magnetic field, the normal to the loop is parallel to the field, so $\theta = 0^\circ$.
Magnetic flux is $\Phi = B A \cos(0^\circ) = 0.5 \times 1 \times 1 = 0.5 Wb$.
2. The magnetic flux linked with a coil (in Wb) is given by the equation $\phi = 5t^2 + 3t + 16$. The magnitude of induced emf in the coil at the fourth second will be: (2020-Covid)
Magnitude of induced emf is $e = |\frac{d\phi}{dt}|$.
Differentiating flux with respect to time: $\frac{d\phi}{dt} = 10t + 3$.
At $t = 4 s$, $e = 10(4) + 3 = 43 V$.
3. A 800 turn coil of effective area $0.05 m^2$ is kept perpendicular to a magnetic field of $5 \times 10^{-5} T$. When the plane of the coil is rotated by $90^\circ$ around any of its coplanar axis in $0.1 s$, the emf induced in the coil will be: (2019)
Initial flux is $\Phi_1 = N B A \cos(0^\circ) = 800 \times (5 \times 10^{-5}) \times 0.05 = 2 \times 10^{-3} Wb$.
After rotating by $90^\circ$, $\Phi_2 = N B A \cos(90^\circ) = 0$.
Magnitude of induced emf is $e = |\frac{\Delta \Phi}{\Delta t}| = \frac{2 \times 10^{-3} - 0}{0.1} = 0.02 V$.
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.