Question

The electric field intensity at points $P$ and $Q$ in the shown arrangement, are in the ratio

• A. $1:2$
• B. $2:1$
• C. $1:1$
• D. $4:3$

Answer 1

The correct option is C $1:1$

Let us consider spherical Gaussian surfaces passing through the points $P$ and $Q$.


Let the area of the Gaussian surface passing through the points $P$ and $Q$ be $A_P$ and $A_Q$ and electric field at the surfaces is $E_P$ and $E_Q$ respectively.

The Gaussian surface passing through $P$ will enclose the charge $q$ only.
$\oint \overrightarrow{E_P}.\overrightarrow{d{A}_P}=\frac{q_{in}}{{ϵ}_0}$
$\Rightarrow E_P{A}_P=\frac{q}{{ϵ}_0}$
$\Rightarrow E_P\left(4\pi r^2\right)=\frac{q}{{ϵ}_0}$
$\therefore E_P=\frac{q}{4\pi {ϵ}_0{r}^2}=\frac{kq}{r^2}$

Similarly, for the Gaussian sphere passing through $Q$,
$q_{in}=q+3q=4q$

Applying Gauss's law gives,
$E_Q{A}_Q=\frac{q_{in}}{{ϵ}_0}=\frac{4q}{{ϵ}_0}$
$\Rightarrow E_Q\left(4\pi \right(2r{)}^2)=\frac{4q}{{ϵ}_0}$
$\Rightarrow E_Q=\frac{4kq}{4r^2}=\frac{kq}{r^2}$

Hence, ratio of electric fields at $P$ and $Q$,
$\frac{E_P}{E_Q}=\frac{\frac{kq}{r^2}}{\frac{kq}{r^2}}$
$\Rightarrow E_P:E_Q=1:1$
Hence, option (c) is correct.