(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by 

$(E_2-E_1)\dot{}\hat{n}\frac{\sigma }{{\epsilon }_0}$
where $\hat{n}$ is a unit vector normal to the surface at a point and $\sigma$ is the surface charge density at that point. (The direction of $\hat{n}$ is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is $\sigma \hat{n}/{ϵ}_0$
(b) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another.  [Hint: For (a), use Gauss's law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.]

(b) A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluium sheet of area $1m^2$. Will he get an electric shock if he touches the metal sheet next morning?

(c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on an average over the globe. Why then does the atmosphere not discharge itself completely in due course and become electrically neutral? In other words, what keeps the atmosphere charged?

(d) What are the forms of energy into which the electrical energy of the atmosphere is dissipated during a lightning?
(Hint: The earth has an electric field of about 100 V$m^{-1}$ at its surface in the downward direction, corresponding to a surface charge density =$-10^{-9}$ C$m^{-2}$. Due to the slight conductivity of the atmosphere up to about 50 km (beyond which it is good conductor), about + 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms and lightning occurring continually allover the globe pump an equal amount of negative charge on the earth.)