$A$ metallic ring of radius $a$ and resistance $R$ is held fixed with its axis along a spatially uniform magnetic field whose magnitude is $B = B_0 \sin \omega t$. Gravity is neglected. Then,

$A$ uniform but time-varying magnetic field $B(t)$ exists in a circular region of radius $a$ and is directed into the plane of the paper,as shown. The magnitude of the induced electric field at point $P$ at a distance $r$ $(r > a)$ from the centre of the circular region is:

$A$ coil is placed in a time-varying magnetic field. The power dissipated due to current induced in the coil is $P_1$. If the number of turns is doubled and the radius of the wire is halved,the power dissipated is $P_2$. Then $P_1: P_2$ is

$A$ square loop of side $2.0\,cm$ is placed inside a long solenoid that has $50$ turns per centimetre and carries a sinusoidally varying current of amplitude $2.5\,A$ and angular frequency $700\,rad\,s^{-1}$. The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is $x \times 10^{-4}\,V$. The value of $x$ is $.........$ (Take $\pi = \frac{22}{7}$)

$A$ uniform magnetic field $B$ exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of $4 \, mm$ and a total length of $30 \, cm$. The magnetic field changes with time at a steady rate $dB/dt = 0.032 \, T s^{-1}$. The induced current in the loop is close to $.... A$ (Resistivity of the metal wire is $1.23 \times 10^{-8} \, \Omega m$).

$A$ rectangular wire loop of sides $8 \; cm$ and $2 \; cm$ with a small cut is moving out of a region of uniform magnetic field of magnitude $0.3 \; T$ directed normal to the loop. Suppose the loop is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of $0.3 \; T$ at the rate of $0.02 \; T \, s^{-1}$. If the cut is joined and the loop has a resistance of $1.6 \; \Omega$,how much power is dissipated by the loop as heat? What is the source of this power?