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Standard XII

Physics

Electric Field Due to a Dipole

Two point cha...

Question

Two point charges $+ q$ and $- 2 q$ are placed at a distance $6 m$ part on a horizontal plane. Find the locus of a point on this plane where the electric potential has zero value.

Circle with radius

4 m

and centre

(0, 0)

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Circle with radius

4 m

and centre

(- 2, 0)

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Circle with radius

2 m

and centre

(- 2, 0)

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Circle with radius

4 m

and centre

(0, - 2)

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Solution

The correct option is B Circle with radius $4 m$ and centre $(- 2, 0)$ .
Let a point $P$ be at a distance $r_{1}$ from point charge $+ q$ , and $r_{2}$ from point charge $- 2 q$ as shown in figure.

Let us assume that the net electric potential $(V_{n e t})$ at $P$ due to charge $+ q$ and $- 2 q$ is zero.
$V_{1} =$ Electric potential at $P$ due to charge $+ q$ .
$\Rightarrow V_{1} = \frac{k q}{r_{1}} \dots (i)$
$V_{2} =$ Electric potential at $P$ due to charge $- 2 q$ .
$\Rightarrow V_{2} = \frac{- 2 k q}{r_{2}} \dots (i i)$

Net electric potential at $P$ is
$V_{n e t} = V_{1} + V_{2} = 0$
Substitute values from equation ( $i$ ) and ( $i i$ )
$\Rightarrow V_{n e t} = \frac{k q}{r_{1}} + \frac{(- 2 k q)}{r_{2}} = 0$
$\Rightarrow k q {\frac{1}{r_{1}} + (\frac{- 2}{r_{2}})} = 0$
$\Rightarrow \frac{1}{r_{1}} - \frac{2}{r_{2}} = 0$
$\Rightarrow r_{2} = 2 r_{1} \dots (i i i)$

It is given that charge $+ q$ is located at the origin and $- 2 q$ is at $x = 6 m$

$\Rightarrow r_{1}^{2} = x^{2} + y^{2}$
and
$r_{2}^{2} = {(6 - x)}^{2} + y^{2}$

Substitute $r_{1}$ and $r_{2}$ in equation ( $i i i$ )

$r_{2}^{2} = 4 r_{1}^{2}$
$\Rightarrow {(6 - x)}^{2} + y^{2} = 4 {x^{2} + y^{2}}$
$\Rightarrow 36 - 12 x + x^{2} + y^{2} = 4 x^{2} + 4 y^{2}$
$\Rightarrow 3 x^{2} + 3 y^{2} + 12 x = 36$
$\Rightarrow x^{2} + y^{2} + 4 x = 12$
$\Rightarrow x^{2} + y^{2} + 4 x + 4 = 12 + 4$
$\Rightarrow x^{2} + 4 x + 4 + y^{2} = 16$
${(x + 2)}^{2} + y^{2} = 16^{2}$

Comparing this equation with the equation of a circle having radius $r$ and centre at $(h, k)$ .

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$
$\Rightarrow r = 4, (h, k) = (- 2, 0)$

So, the locus of point $P$ is a circle with radius $4 m$ and centre $(- 2, 0)$

Hence, option (a) is correct.

$\begin{matrix} Why this question ? Concept: In this question we learn how to calculate the locus of point having zero potential. \end{matrix}$

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