$\sigma$ is the uniform surface charge density of a thin spherical shell of radius $R$. The electric field at any point on the surface of the spherical shell is:

If a charge $q$ is placed on one of the vertices of a cube,then the flux passing through any one face of the cube is . . . . . . .

$A$ solid metallic sphere has a charge $+3 Q$. 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 $B$ $(B > A)$. The electric field at a distance $R$ $(A < R < B)$ from the centre is $(\varepsilon_0 = \text{permittivity of vacuum})$

An electron revolves around an infinite cylindrical wire having a uniform linear charge density of $2 \times 10^{-8} \, C \cdot m^{-1}$ in a circular path under the influence of an attractive electrostatic field,as shown in the figure. The velocity of the electron with which it is revolving is $......... \times 10^6 \, m \cdot s^{-1}$. (Given: mass of electron $= 9 \times 10^{-31} \, kg$)

$A$ 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 $nC/m^2$)

The electric field components in the figure are $E_{x}=\alpha x^{1 / 2}, E_{y}=E_{z}=0,$ in which $\alpha=800 \; N/C \cdot m^{1/2}.$ Calculate
$(a)$ the flux through the cube,and
$(b)$ the charge within the cube. Assume that $a=0.1 \; m$.