A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate $\frac{dB}{dt}$. Loop 2 of radius R is outside the region of magnetic field as shown in the figure. Then the emf generated is-

                  

1. zero in loop 1 and zero in loop 2

2. $-\frac{dB}{dt}{\mathrm{\pi r}}^2$ $\mathrm{in}$ $\mathrm{loop}$ $1$ $\mathrm{and}-\frac{\mathrm{dB}}{\mathrm{dt}}{\mathrm{\pi r}}^2$ $\mathrm{in}$ $\mathrm{loop}$ $2$

3. $-\frac{dB}{dt}{\mathrm{\pi r}}^2$ $\mathrm{in}$ $loop$ $1$ $and$ $zero$ $\mathrm{in}$ $\mathrm{loop}$ $2$

4. $\frac{2dB}{dt}{\mathrm{\pi r}}^2$ $\mathrm{in}$ $loop$ $1$ $and$ $zero$ $\mathrm{in}$ $\mathrm{loop}$ $2$

Two conducting circular loops of radii \(R_1\) and \(R_2\) are placed in the same plane with their centres coinciding. If \(R_1>>R_2\), the mutual inductance \(M\) between them will be directly proportional to:

The magnetic potential energy stored in a certain inductor is \(25~\text{mJ},\) when the current in the inductor is \(60~\text{mA}.\) This inductor is of inductance:
1. \(0.138~\text H\)
2. \(138.88~\text H\)
3. \(1.389~\text H\)
4. \(13.89~\text H\)

A long solenoid of diameter \(0.1\) m has \(2 \times 10^4\) turns per meter. At the center of the solenoid, a coil of \(100\) turns and radius \(0.01\) m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to \(0\) A from \(4\) A in \(0.05\) s. If the resistance of the coil is \(10\pi^2~\Omega\), then the total charge flowing through the coil during this time is:
1. \(16~\mu \text{C}\)
2. \(32~\mu \text{C}\)
3. \(16\pi~\mu \text{C}\)
4. \(32\pi~\mu \text{C}\)

An electron moves on a straight-line path \(XY\) as shown. The \({abcd}\) is a coil adjacent to the path of electrons. What will be the direction of current if any, induced in the coil? 
                       

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. \( \dfrac{1}{x^2} \)
2. \( \dfrac{1}{(2 x-a)^2} \)
3. \( \dfrac{1}{(2 x+a)^2} \)
4. \(\dfrac{1}{(2 x-a)(2 x+a)}\)

A thin semicircular conducting the ring \((PQR)\) of radius \(r\) is falling with its plane vertical in a horizontal magnetic field \(B,\) as shown in the figure. The potential difference developed across the ring when it moves with speed \(v\) is: