$A$ conducting circular loop is placed in a uniform magnetic field,$B = 0.025 \, T$,with its plane perpendicular to the field. 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:

$(a)$ Obtain an expression for the mutual inductance between a long straight wire and a square loop of side $a$ as shown in Figure.
$(b)$ Now assume that the straight wire carries a current of $50\; A$ and the loop is moved to the right with a constant velocity, $v=10\; m / s$. Calculate the induced $emf$ in the loop at the instant when $x=0.2\; m$. Take $a=0.1\; m$ and assume that the loop has a large resistance.

$A$ metal wire of length $2500 \ m$ is kept in east-west direction,at a certain height from the ground. If it falls freely on the ground,then the current induced in the wire when its speed is $10 \ m/s$ is (Resistance of wire $= 25 \ \Omega$,$g = 10 \ m/s^2$ and Earth's horizontal component of magnetic field $B_{H} = 2 \times 10^{-5} \ T$). (in $A$)

Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed $v$ in a uniform magnetic field $B$ directed into the plane of the paper (See figure). If charge densities $\sigma_1$ and $\sigma_2$ are induced on the left and right surfaces,respectively,of the sheet,then (ignore fringe effects):

$A$ long solenoid with $2000$ turns per meter has a small loop of radius $3 \,cm$ placed inside the solenoid normal to its axis. If the current through the solenoid increases steadily from $1.5 \,A$ to $5.5 \,A$ in $\frac{\pi^2}{100} \,s$, the induced emf in the loop is (in $\,mV$)

$A$ conducting rod $PQ$ of length $L = 1.0\, m$ is moving with uniform speed $v = 20\, m/s$ in a uniform magnetic field $B = 4.0\, T$ directed into the paper. $A$ capacitor of capacity $C = 10\, \mu F$ is connected as shown in the figure. Then: