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

each are separated by distance

2 d

. A third charge

+ q

is kept at mid point

O

. Find potential energy of

+ q

as a function of small distance

x

from

O

due to

- q

charges. Sketch P.E.

v / s x

and convince yourself that the charge at

O

is in an unstable equilibrium.

Accepted Answer

$V = \frac{k q}{r}$

Let equilibrium of $O$ is at $P$ at a distance $x$ from mid-point of line joining two charges. Force $O$ on $+ q$ is towards left side and force $F_{B}$ is towards right side, so for equilibrium of $+ q$ at $P$

$F_{A} = F_{B}$

$\frac{- k q^{2}}{(d - x)^{2}} = \frac{- k q^{2}}{(d + x)^{2}}$

$∴ (d - x)^{2} = (d + x)^{2}$

$d - x = d + x$ (Taking square root)

$- 2 x = 0$

$x = 0$

So, Equilibrium position of charge $+ q$ between two $- q$ charges is at mid-point $(O)$ of line joining the two charges $+ q$ and $+ q$ .
Now we have to find out potential energy of $+ q$ as a function of small distance $x$ from balance condition $(O)$ towards any of $(- q)$ charge.
Let new position of charge $(+ q)$ from a small distance $x$ from $(O)$

$U = \frac{k (q) (- q)}{(d - x)} + \frac{k (q) (- q)}{(d + x)} (∵ U = \frac{k q_{1} q_{2}}{r})$

$= - k q^{2} [\frac{1}{(d - x)} + \frac{1}{(d + x)}]$ .

$= - k q^{2} [\frac{d + x + d - x}{(d - x) (d + x)}] = - k q^{2} [\frac{2 d}{d^{2} - x^{2}}]$

$U = \frac{- q^{2}}{4 π ε_{0}} \cdot \frac{2 d}{(d^{2} - x^{2})}$

So, $U$ is the P.E. as a function of $x$

Two charges $- q$ each are separated by distance $2 d$ . A third charge $+ q$ is kept at mid point $O$ . Find potential energy of $+ q$ as a function of small distance $x$ from $O$ due to $- q$ charges. Sketch P.E. $v / s x$ and convince yourself that the charge at $O$ is in an unstable equilibrium.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Two charges -q each are separated by distance 2d. A third charge +q is kept at mid point O. Find potential energy of +q as a function of small distance x from O due to -q charges. Sketch P.E. v/s x and convince yourself that the charge at O is in an unstable equilibrium.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !