Question

(a) Define electric flux. Write its S.I. unit. "Gauss's law in electrostatics is true for any closed surface, no matter what its shape or size is". Justify this statement with the help of a suitable example.
(b) Use Gauss's law to prove that electric field inside a uniformly charged spherical shell is zero.

Answer 1

(a) The electric flux through an area is defined as the electric field multiplied by the area of the surface projected on a plane, perpendicular to the field. Its S.I. unit is voltmeters (Vm) or Newton metres square per coulomb $\left(N{m}^2{C}^{-1}\right)$. The given statement is justified because while measuring the flux, the surface area is more important than its volume or its size.

(b) To prove that the electric field inside a uniformly charged spherical shell is zero, we place a single positive point charge $q$ at the centre of an imaginary spherical surface with radius $R$. The field lines of this point radiate outwards equally in all directions. The magnitude $E$ of the electric field at every point on the surface is given by $E=\frac{1}{4\pi {ϵ}_0}\times \frac{q}{R^2}$.

At each point on the surface, $\overrightarrow{E}$ is $\perp$ to the surface and its magnitude is the same.

Thus, the total electric flux ${\varphi }_E$ is the product of their field magnitude E and the A.

Hence, ${\varphi }_E=EA$

$=\frac{1}{4\pi {ϵ}_0}\frac{q}{R^2}\left(4\pi R^2\right)$

$=\frac{q}{{ϵ}_0}$

If the sphere is uniformly charged, then there is zero charge inside the sphere, according to Gauss's law

When $q=0,{\varphi }_E=\frac{0}{{ϵ}_0}=0$