An electric field is given by $\vec{E} = e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}$. $A$ charge $Q$ is moved by a displacement vector $\vec{r} = a \hat{i} + b \hat{j}$. The work done is:

What is the electric field at the center of a semi-circular ring of radius $R$ having a linear charge density $\lambda$? $\left( k = \frac{1}{4\pi \varepsilon_0} \right)$

Two charges $+4e$ and $+e$ are at a distance $x$ apart. At what distance must a charge $q$ be placed from charge $+e$ so that it is in equilibrium?

Two non-conducting solid spheres of radii $R$ and $2R$,having uniform volume charge densities $\rho_1$ and $\rho_2$ respectively,touch each other. The net electric field at a distance $2R$ from the centre of the smaller sphere,along the line joining the centres of the spheres,is zero. The ratio $\frac{\rho_1}{\rho_2}$ can be;
$(A) -4$ $(B) -\frac{32}{25}$ $(C) \frac{32}{25}$ $(D) 4$

Two point electric charges $+10^{-8} \text{ C}$ and $-10^{-8} \text{ C}$ are placed $0.1 \text{ m}$ apart. Find the magnitude of the total electric field at the centre of the line joining the two charges.

Two particles $A$ and $B$ having charges $20\, \mu C$ and $-5\, \mu C$ respectively are held fixed with a separation of $5\, cm$. At what position should a third charged particle be placed so that it does not experience a net electric force?