(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by $$ \left(E_{2}-E_{1}\right) \cdot \hat{n}=\frac{\sigma}{\epsilon_{0}} $$ 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 $\sigma \hat{n} / \epsilon_{0}$.
(b) Show that the tangential component of the 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.].

Solution:

(a) Allow $E_1$ to represent the electric field on one side of the charged body and $E_2$ to represent the electric field on the opposite side of the charged body for the sake of simplicity. Suppose the infinite planar charged body has a uniform thickness, and the electric field due to one of the charged body’s surfaces is equal to zero.

$\vec{E}{1}=-\frac{\sigma}{2 \epsilon{0}} \hat{n}—(1)$

here,

$\hat{n}=$ unit vector normal to the surface at a point

$\sigma=$ surface charge density at that point

The electric field due to the other surface of the charged body is

$\vec{E}{2}=\frac{\sigma}{2 \epsilon{0}} \hat{n}$

The electric field at any point due to the charge surfaces

$\vec{E}{2}-\vec{E}{1}=\frac{\sigma}{2 \epsilon_{0}} \hat{n}+\frac{\sigma}{2 \epsilon_{0}} \hat{n}=\frac{\sigma}{\epsilon_{0}} \hat{n}$

Since inside the conductor, $\vec{E}_{1}=0$

$$
\vec{E}{2}-\vec{E}{1}=\frac{\sigma}{\epsilon_{0}} \hat{n}–(3)
$$

Therefore, the electric field just outside the conductor is $\frac{\sigma}{\epsilon_{0}} \hat{n}$

(b) Whenever a charged particle is transferred from one location to another in a closed-loop, the work done by the electrostatic field is equal to zero. Thus, the tangential component of the electrostatic field is continuous from one side of a charged surface to the other side of a charged surface.