Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d(d  << I) $ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result charges approach each other with a velocity $v$. Then as a function of distance $x$ between them,

An electric dipole is placed along the $x$ -axis at the origin $O.$ A point $P$ is at a  distance of $20\, cm$ from this origin such that $OP$ makes an angle $\frac{\pi}{3}$ with  the $x$ -axis. If the electric field at $P$ makes an angle $\theta$ with the $x$ -axis, the value of $\theta$ would be

A parallel plate capacitor is charged to a potential difference of $100\,V$ and disconnected from the source of $emf$. A slab of dielectric is then inserted between the plates. Which of the following three quantities change?

$(i)$ The potential difference    $(ii)$ The capacitance    $(iii)$ The charge on the plates

In the figure a potential of $+ 1200\, V$ is given to point $A$ and point $B$ is earthed, what is the potential at the point $P$….$V$

In steady state heat conduction, the equations that determine the heat current $j ( r )$ [heat flowing per unit time per unit area] and temperature $T( r )$ in space are exactly the same as those governing the electric field $E ( r )$ and electrostatic potential $V( r )$ with the equivalence given in the table below.

We exploit this equivalence to predict the rate $Q$ of total heat flowing by conduction from the surfaces of spheres of varying radii, all maintained at the same temperature. If $\dot{Q} \propto R^{n}$, where $R$ is the radius, then the value of $n$ is