The region between two concentric spheres of radii $$'a'$$ and $$'b',$$ respectively (see figure), have volume charge density $$\rho = {A \over r},$$ where $$A$$ is a constant and $$r$$ is the distance from the center. A such that the electric field in the region between the spheres will be constant, is :

A long cylindrical shell carries positives surfaces change $$\sigma $$ in the upper half and negative surface charge - $$\sigma $$ in the lower half. The electric field lines around the cylinder will look like figure given in :
(figures are schematic and not drawn to scale)

Assume that an electric field $$\overrightarrow E = 30{x^2}\widehat i$$ exists in space. Then the potential difference $${V_A} - {V_O},$$ where $${V_O}$$ is the potential at the origin and $${V_A}$$ the potential at $$x=2$$ $$m$$ is :

Two charges, each equals to $$q,$$ are kept at $$x=-a$$ and $$x=a$$ on the $$x$$-axis. A particle of mass $$m$$ and charge $${q_0} = {q \over 2}$$ is placed at the origin. If charge $${q_0}$$ is given a small displacement $$\left( {y < < a} \right)$$ along the $$y$$-axis, the net force acting on the particle is proportional to