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

The alternating e.m.f. induced in the secondary coil of a transformer is mainly due to

$A$ 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?

$A$ 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.

$A$ long circular tube of length $10 \ m$ and radius $0.3 \ m$ carries a current $I$ along its curved surface as shown. $A$ wire-loop of resistance $0.005 \ \Omega$ and of radius $0.1 \ m$ is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $I = I_0 \cos(300t)$ where $I_0$ is constant. If the magnetic moment of the loop is $N \mu_0 I_0 \sin(300t)$,then $N$ is:

Two concentric circular coils,$C_{1}$ and $C_{2}$,are placed in the $XY$ plane. $C_{1}$ has $500$ turns and a radius of $1\; cm$. $C_{2}$ has $200$ turns and a radius of $20\; cm$. $C_{2}$ carries a time-dependent current $I(t) = (5t^{2} - 2t + 3)\; A$,where $t$ is in $s$. The $emf$ induced in $C_{1}$ (in $mV$) at the instant $t = 1\; s$ is $\frac{4}{x}$. The value of $x$ is: