A conducting ring of radius r is placed in a varying magnetic field perpendicular to the plane of the ring. If the rate at which the magnetic field varies is x, the electric field intensity at any point of the ring is ________.

The correct option is rx.

To understand why this is the case, we can analyze the relationship between the induced electric field and the varying magnetic field using Faraday's law of electromagnetic induction.

Faraday's law states that a time-varying magnetic field induces an electromotive force (EMF) and consequently an induced electric field. The induced electromotive force $e$ around a closed loop is given by the rate of change of magnetic flux $\Phi$ through the loop:

$e=-\frac{d\Phi }{dt}$

The magnetic flux $\Phi$ through the area of the ring of radius $r$ is given by the product of the magnetic field $B$ and the area $A$:

$\Phi =B·A$

Here, the rate at which the magnetic field varies is given as:
$\frac{dB}{dt}=x$

The induced electric field intensity $E$ at any point on the conducting ring is directly proportional to the radius of the ring $r$ and the rate of variation of the magnetic field $x$. This gives the relation:

$E=rx$

Therefore, the electric field intensity at any point of the ring is rx.