Name: Equipotential Surfaces and Its Properties
Uploaded: 2019-07-19
Duration: 7 min 22 s
Description: The surface over which the electric potential is the same everywhere is called an equipotential surface. Let's more about it and its properties in this video.

Question

Two charges $q_{1}$ and $q_{2}$ are placed at $(0, 0, d)$ and $(0, 0, - d)$ respectively. Find locus of points where the potential a zero.

Accepted Answer

Let the potential at any point $P (x, y, z)$ is zero then $V_{1} + V_{2} = 0$

$\frac{k q_{1}}{\sqrt{x^{2} + y^{2} + (z - d)^{2}}} + \frac{k q_{2}}{\sqrt{x^{2} + y^{2} + (z + d)^{2}}} = 0$

$\frac{q_{1}}{\sqrt{x^{2} + y^{2} + (z - d)^{2}}} + \frac{q_{2}}{\sqrt{x^{2} + y^{2} + (z + d)^{2}}} = 0$

$(0, 0, d)$

$\frac{q_{1}}{q_{2}} = \frac{- \sqrt{x^{2} + y^{2} + (z - d)^{2}}}{\sqrt{x^{2} + y^{2} + (z + d)^{2}}}$

$\frac{q_{1}^{2}}{q_{2}^{2}} = \frac{x^{2} + y^{2} + z^{2} + d^{2} - 2 z d}{x^{2} + y^{2} + z^{2} + d^{2} + 2 z d} . . . . . . (1)$

Taking Componendo and dividend

We know if $\frac{x}{a} = \frac{y}{b}$ then $\frac{x + a}{x - a} = \frac{y + b}{y - b}$

Therefore from $(1)$

$\frac{{(\frac{q_{1}}{q_{2}})}^{2}}{1} = \frac{x^{2} + y^{2} + z^{2} + d^{2} - 2 z d}{x^{2} + y^{2} + z^{2} + d^{2} + 2 d z}$

$\frac{{(\frac{q_{1}}{q_{2}})}^{2} + 1}{{(\frac{q_{1}}{q_{2}})}^{2} - 1} = \frac{x^{2} + y^{2} + z^{2} + d^{2} - 2 d z + (x^{2} + y^{2} + z^{2} + d^{2} + 2 d z)}{x^{2} + y^{2} + z^{2} + d^{2} - 2 d z - (x^{2} + y^{2} + z^{2} + d^{2} + 2 d z)}$

$[\frac{{(\frac{q_{1}}{q_{2}})}^{2} + 1}{{(\frac{q_{1}}{q_{2}})}^{2} - 1}] = \frac{2 (x^{2} + y^{2} + z^{2} + d^{2})}{- 4 d z}$

$x^{2} + y^{2} + z^{2} + d^{2} = - 2 d z [\frac{{(\frac{q_{1}}{q_{2}})}^{2} + 1}{{(\frac{q_{1}}{q_{2}})}^{2} - 1}]$

$x^{2} + y^{2} + z^{2} + 2 d [\frac{{(\frac{q_{1}}{q_{2}})}^{2} + 1}{{(\frac{q_{1}}{q_{2}})}^{2} - 1}] z + d^{2} = 0$

$x^{2} + y^{2} + z^{2} + 2 d [\frac{q_{1}^{2} + q_{2}^{2}}{q_{1}^{2} - q_{2}^{2}}] z + d^{2} = 0$

This is the equation of sphere

Two charges $q_{1}$ and $q_{2}$ are placed at $(0, 0, d)$ and $(0, 0, - d)$ respectively. Find locus of points where the potential a zero.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

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.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Two charges $q_{1}$ and $q_{2}$ are placed at $(0, 0, d)$ and $(0, 0, - d)$ respectively. Find locus of points where the potential a zero.