Question

Two charges q₁ and q₂ are placed at (0, 0, d) and (0, 0, – d) respectively. Find locus of points where the potential a zero.

Answer 1

Following the principle of superposition of potentials as
described in last section, let us find the potential V due to a collection of discrete point charges q₁, q₂, …, q_n, at a point P.

The potential at P due to the system of point charges is given as the sum of their individual potentials at P, V = `1/(4piε_0) sum q_i/r_i`

As we know, the potential at point P is V = `sumV_i`,

 Where `V_i = q_i/(4piε_0) ; r_i` = magnitude of position vector P relative to q_r

Then `V = 1/(4piε_0) sum q_i/r_"pi"`

Let us take a point on the required plane as (x, y, z). The two charges lies on z-axis at a separation of 2d. The potential at the point P due to two charge is given by

`q_1/(sqrt(x^2 + y^2 + (z - d)^2)) + q_2/(sqrt(x^2 + y^2 + (z + d)^2))` = 0

∴ `q_1/(sqrt(x^2 + y^2 + (z - d)^2)) = (-q_2)/(sqrt(x^2 + y^2 + (z + d)^2))`

On squaring and simplifying, we get

`x^2 + y^2 + z^2 + [((q_1/q_2)^2 + 1)/((q_1/q_2)^2 - 1)] (2zd) + d^2 - 0`

The standard equation of sphere is `x^2 + y^2 + z^2 + 2ux + 2uy + 2wz + g` = 0

With centre `(-u, -v, -w)` and radius `sqrt(u^2 + v^2 + w^2) - g`

Hence centre of sphere will be `(0, 0 - d[(q_1^2 + q_2^2)/(q_1^2 - q_2^2)])`

And radius is `r = sqrt((d[(q_1^2 + q_2^2)/(q_1^2 - q_2^2)])^2 - d^2) = (2q_1q_2d)/(q_1^2 - q_2^2)`