A series combination of $n_1$ capacitors, each of value $C_1$, is charged by a source of potential difference $4\,V$. When another parallel combination $n_2$ capacitors, each of value $C_2$, is charged by a source of potential difference $V$, it has the same (total) energy store in it, as the first combination has. The value of $C_2$, in terms of $C_1$, is then

Charge $q$ is uniformly distributed over a thin half ring of radius $R$. The electric field at the centre of the ring is

Two spherical conductors $B$ and $C$ having equal radii and carrying equal charges in  them repel each other with a force $F$ when kept apart at some some distance. $A$ third  spherical conductor having same radius as that of $B$ but uncharged, is brought in contact with $B$, then brought in contact with $C$ and finally removed away from both.  The new force of repulsion between $B$ and $C$ is-

Two point charges $+8q$ and $-2q$ are located at $x = 0$ and $x = L$ respectively. The location of a point on the $x-$ axis at which the net electric field due to these two point charges is zero is

A thin square plate is placed in $x-y$ plane as shown in fig. such that is centre coinsides with origine it's charge density at point $(x, y)$ is $\sigma  = \sigma _0xy$ (where $\sigma _0$ is constant). Find total charge on the plate.