Name: Dipole in an Electric Field
Uploaded: 2019-07-19
Duration: 11 min 59 s
Description: Dipoles exists everywhere. They apply to both electricity and magnetism in physics. Here in this video, let us explore how an electric dipole interacts with...

Question

What do you mean by the potential energy of a system of charges? Derive energy expression for two point charges separated by a distance. Generalise the result for a system of

n

point charges.

Accepted Answer

The amount of work done in bringing any amount of charge from infinity at any point in electric field against the electrostatic force ,is called electric potential potential energy due to system of charges.

Expression coulombs law, $F = \frac{1}{4 {πε}_{0}} \cdot \frac{q_{1} q_{2}}{x^{2}}$ ,along $\vec{op}$

Total work done in bringing $q_{0}$ from $\infty$ at point p is

$w = \int_{\infty}^{r} - F \cdot dx = \int_{\infty}^{r} \frac{1}{4 {πε}_{0}} \cdot \frac{q_{1} q_{2}}{x^{2}} dx$

$= \frac{- q_{1} q_{2}}{4 {πε}_{0}} \int_{\infty}^{r} x^{- 2} dx = \frac{- 2, q_{2}}{4 {πε}_{0}} {[\frac{x^{- 2 + 1}}{- 2 + 1}]}_{\infty}^{r}$

$= \frac{q_{1} q_{2}}{4 {πε}_{0}} {[\frac{1}{x}]}_{\infty}^{r} = \frac{q_{1} q_{2}}{4 {πε}_{0}} [\frac{1}{r} - \frac{1}{\infty}]$

$∴ w = \frac{q_{1} q_{2}}{4 {πε}_{0}} . \frac{1}{r} [\frac{1}{\infty} = 0]$

this work done is called potential energy due to system of two charges $q_{1}$ and $q_{2}$

$U = \frac{1}{4 {πε}_{0}} \frac{q_{1} q_{2}}{r}$

Electric potential energy due to system of n-charge

$U = U_{1} + U_{2} + U_{3} + \dots + U_{n}$

$= \frac{1}{4 {πε}_{0}} [\frac{q_{1} q_{2}}{r_{12}} + \frac{q_{2} q_{3}}{r_{23}} + \frac{q_{3} q_{1}}{r_{31}} + . . . . . . . . . . i terms]$

$[U = \frac{1}{2} \cdot (\frac{1}{4 {πε}_{0}}) \sum_{i^{'} + 1}^{n} \sum_{j = 1}^{n} \frac{q_{i} q_{j}}{r_{ij}}] (i \neq j)$

the double sum no notation $\sum_{i^{} = 1}^{n} \sum_{j = 1}^{n}$

says :- first we take $i = 1$ sum over

$j = 2, 3, 4, \dots, n$ then take $i = 2$

and sum over $j = 1, 3, 4, \dots n$

then $i = 3$ and sum over $j = 1, 2, 4, \dots, n$

and so on up to $i = n$ and $j = 1, 2, 3, \dots (n - 1)$

clearly this includes every pair and correct for the we have put the factor $\frac{1}{2}$ .

What do you mean by the potential energy of a system of charges? Derive energy expression for two point charges separated by a distance. Generalise the result for a system of $n$ point charges.

Important Questions on Electric Potential

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