Electric Potential Energy of a System of two Point Charges

 Electric Potential Energy of a System of two Point Charges

    The electric potential energy of a system of two-point charges is the energy required to bring the charges together or move them apart against the electrostatic force, and it is equal to the work done by an external force.

    Consider two charges Q and q separated by a distance `r_1`. If the charge q is moved along the line joining the charge and the final separation becomes `r_2`, then the work done by the electric force during the process is

Electric Potential Energy of a System of two Point Charges

`W_{ex} = \int_{r_1}^{r_2} \frac{K Q q}{r^2} dr`

`W_{ex} = K Q q \int_{r_1}^{r_2} \frac{1}{r^2} dr`

`W_{ex} = K Q q \int_{r_1}^{r_2} r^{- 2} dr`

`W_{ex} = K Q q [\frac{r^{-2+1}}{-2+1}]_{r_1}^{r_2}`

`W_{ex} = K Q q [\frac{r^{-1}}{-1}]_{r_1}^{r_2}`

`W_{ex} = - K Q q [\frac{1}{r}]_{r_1}^{r_2}`

`W_{ex} = - K Q q [\frac{1}{r_2} - \frac{1}{r_1}]`

`U_1 - U_2 = - K Q q [\frac{1}{r_2} - \frac{1}{r_1}]`

We know that `U_1` = 0 for r = `r_1 = \infty`

`0 - U_r = - K Q q [\frac{1}{r_2} - \frac{1}{\infty}]`        `(\because U_2 = U_r)`

`0 - U_r = - K Q q [\frac{1}{r} - 0]`        `(\because \frac{1}{\infty} = 0 \text{and} r_2 = r)`

` - U_r = - K Q q [\frac{1}{r}]`

` - U_r = - \frac{K Q q}{r}`

` U_r =  \frac{K Q q}{r}`

Putting `Q = q_1` and `q = q_2`

` U_r =  \frac{K q_1 q_2}{r}`

    If the two charges are of the same sign (like charges), then the work done by the external force will be positive because this work is against their mutual repulsion. If the work is stored in the form of the potential energy of the system, then the potential energy will be positive. If the system is free, then the potential energy due to the repulsion of charges from each other will transform into kinetic energy of the charges. 

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