$A$ solenoid of radius $R$ and length $L$ has a current $I = I_0 \sin \omega t$. The value of the induced electric field at a distance $r$ inside the solenoid is:

$A$ long straight solenoid with a cross-sectional radius $a$ and number of turns per unit length $n$ has a current varying with time as $I = I_0 \sin(\omega t)$ (or simply $dI/dt = I$). The magnitude of the electric field as a function of distance $r$ from the solenoid axis is,

$A$ 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 in the figure. The magnitude of the induced electric field at a point '$P$' at a distance '$r$' $(r > a)$ from the center of the circular region 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:

Is relative motion an absolute condition for inducing $emf$?

$A$ 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.