$A$ metal disc of radius $R$ rotates with an angular velocity $\omega$ about an axis perpendicular to its plane passing through its centre in a magnetic field of induction $B$ acting perpendicular to the plane of the disc. The induced e.m.f. between the rim and axis of the disc is (magnitude only):

$(a)$ 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.

$A$ jet plane is travelling towards west at a speed of $1800\; km/h$. What is the voltage difference developed between the ends of the wing having a span of $25\; m$,if the Earth's magnetic field at the location has a magnitude of $5 \times 10^{-4}\; T$ and the dip angle is $30^{\circ}$ (in $; V$)?

Explain the motional $emf$ by the Lorentz force acting on the free charge carriers of a conductor.

$A$ straight conductor of length $0.6 \, m$ is moved with a speed of $10 \, ms^{-1}$ perpendicular to a magnetic field of induction $1.2 \, Wb \cdot m^{-2}$. The induced e.m.f. across the conductor is (in $V$)

$A$ rod of $10 \ cm$ length is moving perpendicular to a uniform magnetic field of intensity $5 \times 10^{-4} \ Wb/m^2$. If the acceleration of the rod is $5 \ m/s^2$, then the rate of increase of induced $emf$ is . . . . . . .