A time varying magnetic flux passing through a coil is given by $\varphi =x{t}^2$, if at $t=3 s$, the emf induced is $9 V$, then the value of $x$ is

A uniform magnetic field exists along $x$-axis, find the flux through a square loop of side $a$ inclined at $60°$ to $(+x)$ axis as shown in figure.

An infinitely long wire carries current $i_0$ along $+z$ axis. Find flux through the right angled isosceles triangular region coplanar with the wire as shown in figure.

The loop moves towards a region of magnetic field. Draw graph of $i(+$ ve direction is marked in figure $)$ and time. Speed of coil is constant and equal to $v .$.

The loop shown in figure is rotating with angular velocity $\omega$ around a perpendicular axis passing through $O$. Which of the following graph correctly represent emf developed in the loop as function of $\theta$ ? ( $\theta$ is shown in diagram) (Consider anti-clockwise current due to $+ ve$ emf)

A rod is moving on rails as shown in figure. Outside magnetic field is given by $B_0$. Resistance of system is $R$ and length is $l .$

(a) Find direction of induced current.
(b) Find emf and current.
(c) Find magnetic force on the rod.

The wire is shaped like a parabola $y^2=4ax$, and the magnetic field is uniform and out of the plane of the paper perpendicularly. At the instant the rod crosses the point $(a, 0)$. Induced emf in the rod is