$A$ thin square plate is placed in the $x-y$ plane as shown in the figure,such that its center coincides with the origin. Its charge density at point $(x, y)$ is $\sigma = \sigma_0 xy$ (where $\sigma_0$ is a constant). Find the total charge on the plate.

Two spherical hollow spheres of radii $R_1$ and $R_2$ are charged with the same charge $Q$. If $\sigma_1$ and $\sigma_2$ are their respective surface charge densities, then the ratio $\sigma_1 : \sigma_2$ is:

Obtain the formula for the electric field due to a long thin wire of uniform linear charge density $\lambda$ without using Gauss's law.

Find the surface charge density at the intersection of the plane $x = 3 \, m$ and the $x$-axis,given a uniform line charge of $8 \, nC/m$ lying along the $z$-axis in free space.

$A$ point charge $q = 1 \mu C$ is located at a distance $2 \text{ cm}$ from one end of a thin insulating wire of length $L = 10 \text{ cm}$ having a charge $Q = 24 \mu C$, distributed uniformly along its length, as shown in the figure. The force between $q$ and the wire is . . . . . . $N$. (Use $\frac{1}{4 \pi \epsilon_0} = 9 \times 10^9 \text{ N} \cdot \text{m}^2 / \text{C}^2$)

$A$ particle of mass $2 \times 10^{-6} \ kg$ with a charge $5 \times 10^{-6} \ C$ is hanging in air above a similarly charged conducting surface. The charge density of the surface is (Assume $\epsilon_0 = 8.85 \times 10^{-12} \ C^2 N^{-1} m^{-2}$ and $g = 10 \ m \ s^{-2}$)