The electric potential in a region around the origin is given by $V(x) = 4x^2$. The total charge enclosed in a cube of side $1 \ m$ centered at the origin is (in Coulombs) ...

Charges $Q, 2Q$ and $4Q$ are uniformly distributed in three dielectric solid spheres $1, 2$ and $3$ of radii $R/2, R$ and $2R$ respectively,as shown in the figure. If magnitudes of the electric fields at point $P$ at a distance $R$ from the centre of spheres $1, 2$ and $3$ are $E_1, E_2$ and $E_3$ respectively,then:

The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about $150\, N/C$,directed inward towards the center of the Earth. This gives the total net surface charge carried by the Earth to be......$kC$ [Given ${\varepsilon _0} = 8.85 \times {10^{ - 12}}\,{C^2}/(N \cdot m^2), {R_E} = 6.37 \times {10^6}\,m$]

$A$ hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is $(\sigma / 2 \varepsilon_{0}) \hat{n}$,where $\hat{n}$ is the unit vector in the outward normal direction,and $\sigma$ is the surface charge density near the hole.

An electron is projected towards a metal plate with a surface charge density $\sigma = -2 \times 10^{-6} \ C/m^2$. If the electron strikes the plate with zero velocity,calculate the distance $d$ between the electron gun and the plate in terms of its initial kinetic energy $E_k$.

$A$ solid metallic sphere has a charge $+3Q$. Concentric with this sphere is a conducting spherical shell having charge $-Q$. The radius of the sphere is $a$ and that of the spherical shell is $b$ $(b > a)$. What is the electric field at a distance $R$ $(a < R < b)$ from the centre?