$A$ uniform but time-varying magnetic field is present in a circular region of radius $R$. The magnetic field is perpendicular and into the plane of the loop and the magnitude of the field is increasing at a constant rate $\alpha$. There is a straight conducting rod of length $2R$ placed as shown in the figure. The magnitude of the induced emf across the rod is:

$A$ point charge $Q$ is moving in a circular orbit of radius $R$ in the $x$-$y$ plane with an angular velocity $\omega$. This can be considered as equivalent to a loop carrying a steady current $I = \frac{Q\omega}{2\pi}$. $A$ uniform magnetic field along the positive $z$-axis is now switched on,which increases at a constant rate from $0$ to $B$ in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that,for an orbiting charge,the magnetic dipole moment is proportional to the angular momentum with a proportionality constant $\gamma$.
$1.$ The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is:
$(A)$ $\frac{BR}{4}$ $(B)$ $\frac{BR}{2}$ $(C)$ $BR$ $(D)$ $2BR$
$2.$ The change in the magnetic dipole moment associated with the orbit,at the end of the time interval of the magnetic field change,is:
$(A)$ $-\gamma BQR^2$ $(B)$ $-\gamma \frac{BQR^2}{2}$ $(C)$ $\gamma \frac{BQR^2}{2}$ $(D)$ $\gamma BQR^2$
Give the answer for question $1$ and $2$.

In the diagram shown,a time-varying non-uniform magnetic field passes through a circular region of radius $R$. The magnetic field is directed outwards and it is a function of radial distance $r$ and time $t$ according to the relation $B = B_0rt$. What is the induced electric field strength at a radial distance $R/2$ from the center?

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

$A$ magnetic field in a certain region is given by $\vec B = B_0 \cos(\omega t) \hat k$. $A$ coil of radius $a$ with resistance $R$ is placed in the $xy$-plane with its centre at the origin in the magnetic field (see figure). Find the magnitude and the direction of the current at $(a, 0, 0)$ at $t = \frac{\pi}{2\omega}$,$t = \frac{\pi}{\omega}$,and $t = \frac{3\pi}{2\omega}$.

$A$ 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: