Consider a solid insulating sphere of radius $R$ with charge density varying as $\rho = \rho _0r^2$ ($\rho _0$ is a constant and $r$ is measure from centre). Consider two points $A$ and $B$ at distance $x$ and $y$ respectively $(x < R, y > R)$ from the centre. If magnitudes of electric fields at points $A$ and $B$ are equal, then
${x^2}y = {R^3}$
${x^3}y^2 = {R^5}$
${x^2}y^3 = {R^5}$
$\frac{{{x^4}}}{y} = {R^5}$
The plates of a parallel plate capacitor are charged up to $100\, volt$. A $2\, mm$ thick plate is inserted between the plates, then to maintain the same potential difference, the distance between the capacitor plates is increased by $1.6\, mm$. The dielectric constant of the plate is
If on the concentric hollow spheres of radii $r$ and $R( > r)$ the charge $Q$ is distributed such that their surface densities are same then the potential at their common centre is
A network of four capacitors of capacity equal to $C_1 = C, C_2 = 2C, C_3 = 3C$ and $C_4=4C$ are conducted to a battery as shown in the figure. The ratio of the charges on $C_2$ and $C_4$ is
The potential $V$ is varying with $x$ and $y$ as $V = \frac{1}{2}({y^2} - 4x)\,volts$ The field at $(1\,m,\,1\,m)$ is
A charge $Q$ is placed at each of the opposite corners of a square. A charge $q$ is placed at each of the other two corners. If the electrical force on $Q$ is zero, then $Q/q$ equals