Let $\rho (r) = \frac{Q}{\pi R^4} r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$. For a point $p$ inside the sphere at distance $r_1$ from the centre of the sphere,the magnitude of the electric field is:

Obtain the expression for the electric field due to a uniformly charged spherical shell at a point outside it.

In finding the electric field using Gauss's Law,the formula $|\overrightarrow{E}| = \frac{q_{enc}}{\varepsilon_{0}|A|}$ is applicable. In the formula,$\varepsilon_{0}$ is the permittivity of free space,$A$ is the area of the Gaussian surface,and $q_{enc}$ is the charge enclosed by the Gaussian surface. The equation can be used in which of the following situations?

$A$ small bob of mass $100 \ mg$ and charge $+10 \ \mu C$ is connected to an insulating string of length $1 \ m$. It is brought near to an infinitely long nonconducting sheet of charge density $\sigma$ as shown in the figure. If the string subtends an angle of $45^{\circ}$ with the sheet at equilibrium,the charge density of the sheet will be (Given,$\varepsilon_0 = 8.85 \times 10^{-12} \ F/m$ and acceleration due to gravity,$g = 10 \ m/s^2$): (in $nC/m^2$)

$A$ long, straight wire is surrounded by a hollow, thin, long metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of $\lambda$, and the cylinder has a net charge per unit length of $2\lambda$. The radius of the cylinder is $R$. Which of the following statements is correct?

$A$ conducting sphere of radius $0.1 \ m$ has a uniform charge density $1.8 \ \mu C/m^2$ on its surface. The electric field in free space at a radial distance of $0.2 \ m$ from the center of the sphere is $(\varepsilon_0 = \text{permittivity of free space})$