$A$ uniformly charged thin spherical shell of radius '$R$' has a uniform surface charge density '$\sigma$'. It is made of two identical hemispherical shells held together by pressing them with a force '$F$' as shown. '$F$' is proportional to [$\epsilon_0$ = permittivity of free space].

$A$ non-conducting solid sphere of radius $R$ is uniformly charged. The magnitude of the electric field due to the sphere at a distance $r$ from its center is:
$(1) \text{ Increases with increase in } r \text{ for } r < R$
$(2) \text{ Decreases with increase in } r \text{ for } 0 < r < \infty$
$(3) \text{ Decreases with increase in } r \text{ for } R < r < \infty$
$(4) \text{ Is continuous at } r = R$

$A$ spherically symmetric charge distribution is characterized by a charge density $\rho(r) = \rho_0 \left( \frac{5}{4} - \frac{r}{R} \right)$ for $r \le R$ and $\rho(r) = 0$ for $r > R$,where $r$ is the distance from the origin. The electric field at a distance $r$ from the origin $(r < R)$ is given by:

Obtain Coulomb's law from Gauss's law.

$A$ 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 cylindrical volume contains a uniformly distributed charge of density $\rho \; C m^{-3}$. The electric field inside the cylindrical volume at a distance $x = \frac{2 \varepsilon_{0}}{\rho} \; m$ from its axis is $....... V m^{-1}$.