Let a total charge $2Q$ be distributed in a sphere of radius $R$, with the charge density given by $\rho(r) = kr$, where $r$ is the distance from the centre. Two charges $A$ and $B$, of $-Q$ each, are placed on diametrically opposite points, at equal distance, $a$, from the centre. If $A$ and $B$ do not experience any force, then

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin $O$. In this distribution, a spherical cavity of radius $R_2$, centred at $P$ with distance $O P=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\overrightarrow{ r }$ is $\overrightarrow{ E }(\overrightarrow{ r })$, then the correct statement$(s)$ is(are) $Image$

Let there be a spherically symmetric charge distribution with charge density varying as $\rho (r)=\;\rho _0\left( {\frac{5}{4} - \frac{r}{R}} \right)$, upto $r = R$ ,and $\rho (r) = 0$ for $r > R$ , where $r$ is the distance from the origin. The electric field at a distance $r(r < R)$ from the origin is given by

The electric field at $20 \,cm$ from the centre of a uniformly charged non-conducting sphere of radius $10 \,cm$ is $E$. Then at a distance $5 \,cm$ from the centre it will be

The electric field at a distance $r$ from the centre in the space between two concentric metallic spherical shells of radii $r_1$ and $r_2$ carrying charge $Q_1$ and $Q_2$ is $(r_1 < r < r_2)$

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho $. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2R$ from the axis of the cylinder, is given by the expression $\frac{{23\rho R}}{{16K{\varepsilon _0}}}$ .The value of $K$ is