Question

Consider a thin spherical shell of radius $R$ consisting of uniform surface charge density $\sigma$. The electric field at a point of distance $x$ from its centre and outside the shell is

• A. inversely proportional to $\sigma$
• B. directly proportional to $x^2$
• C. directly proportional to $R$
• D. inversely proportional to $x^2$

Answer 1

The correct option is D inversely proportional to $x^2$

$\text{Step 1: Electric field outside the spherical shell}$

For a thin uniformly charged spherical shell, the field points outside the shell at a distance $x$ from the center is same as that of point charge placed at its center, Given by:

$E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{x^2}$ $....\left(1\right)$

$\text{Step 2: Charge on the spherical shell}$

We know that the charge on spherical shell can be given as :

$Q=\text{Surface charge density}\times \text{Area of spherical shell}$

$\Rightarrow Q=\sigma \times 4\pi R^2$

So, Equation $\left(1\right)$ becomes

$\therefore E=\frac{1}{4\pi {\epsilon }_0}\frac{\sigma 4\pi R^2}{x^2}=\frac{\sigma R^2}{{{\epsilon }_{0}x}^2}$

Hence, $E$ is inversely proportional to square of the distance from the centre$\left(x\right)$.

Therefore, Option(D) is correct.