The electric field due to an infinitely long thin straight wire with uniform linear charge density of $2.5 \times 10^{-7} \ Cm^{-1}$ at a radial distance of $x$ from the wire is $7.5 \times 10^4 \ NC^{-1}$. Then $x=$ (in $cm$)

The electric field intensity on the surface of a solid charged sphere of radius $r$ and volume charge density $\rho$ is ( $\epsilon_0=$ permittivity of free space).

Two concentric conducting thin spherical shells $A$ and $B$ having radii $r_A$ and $r_B$ $(r_B > r_A)$ are charged to $Q_A$ and $-Q_B$ $(|Q_B| > |Q_A|)$. The electric field along a line passing through the centre is:

The electric intensity outside a charged sphere of radius $R$ at a distance $r$ $(r > R)$ is

Consider a sphere of radius $R$ with charge density distributed as:
$\rho(r) = kr$ for $r \leq R$
$\rho(r) = 0$ for $r > R$
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where $e$ is the elementary charge. Where can two protons be embedded such that the force on each of them is zero? Assume that the introduction of the protons does not alter the charge distribution.

At a distance $l$ from a uniformly charged long wire, a charged particle is thrown radially outward with a velocity $u$ in the direction perpendicular to the wire. When the particle reaches a distance $2l$ from the wire, its speed is found to be $\sqrt{2}u$. The magnitude of the velocity, when it is a distance $4l$ away from the wire is (ignore gravity)