Show via the construction of a suitable Gaussian surface, that the capacitance of a spherical capacitor consisting of two concentric shells of radii $$a$$ and $$b$$ is given by $$C = 4\pi {\varepsilon _0}{{ab} \over {\left( {b - a} \right)}}$$ where $${\varepsilon _0}$$ is the free space permittivity.

A charge $$+Q$$ is uniformly distributed throughout the volume of a dielectric sphere of radius $$R$$ and dielectric constant $$'{\varepsilon _R}'.$$ Based on Gauss law, determine the expressions for the electric field $$E$$ as a function of distance $$'r'$$ from the center of the sphere, within the ranges $$0 < r < R$$ and $$R \le r.$$ Indicate expression $$(s)$$ for the critical point $$(s)$$ on the sketch.

An infinite number of charges, each equal to $$'q'$$ are placed along the $$x=1,x=2, x=4, x=8, x=16$$ and so on, Find the potential and electric field at point $$x=0,$$ due to these system of charges.

Determine the electric field intensity at the point $$P$$ for the arrangement shown in Fig.