$A$ uniform magnetic field $B$ exists in a cylindrical region of radius $R = 10 \, cm$ as shown in the figure. $A$ uniform wire of length $L = 80 \, cm$ and resistance $R_{wire} = 4.0 \, \Omega$ is bent into a square frame of side length $a = 20 \, cm$ and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of $\frac{dB}{dt} = 0.010 \, T/s$,find the current induced in the frame.

$A$ 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?

At time $t=0$, a magnetic field of $1000 \; \text{Gauss}$ is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to $500 \; \text{Gauss}$ in the next $5 \; \text{s}$, then the induced $EMF$ in the loop is ........ $\mu \text{V}$.

$A$ magnetic field $\vec{B} = B_0 \sin(\omega t) \hat{k}$ covers a large region where a wire $AB$ slides smoothly over two parallel conductors separated by a distance $d$ as shown in the figure. The wires are in the $xy$-plane. The wire $AB$ (of length $d$) has resistance $R$ and the parallel wires have negligible resistance. If $AB$ is moving with velocity $v$,what is the current in the circuit? What is the force needed to keep the wire moving at constant velocity?

$A$ non-conducting ring (of mass $m$,radius $r$,having charge $Q$) is placed on a rough horizontal surface in a region with a transverse magnetic field. The magnetic field is increasing with time at a rate $R = dB/dt$. If the coefficient of friction between the surface and the ring is $\mu$,for the ring to remain in equilibrium,$\mu$ should be greater than:

$A$ square loop of side $2 \text{ cm}$ is placed in a time-varying magnetic field with magnitude $B = 0.4 \sin(300t) \text{ T}$. The normal to the plane of the loop makes an angle of $60^{\circ}$ with the field. The maximum induced emf produced in the loop is . . . . . . $\text{mV}$.