Explain electrostatic potential energy difference and give the noteworthy comments on it.

At every point in electric field a particle with charge $q$ possesses a certain electrostatic potential energy this work done increases its potential energy by an amount equal to the potential energy difference between points $R$ and $P$.

Thus, potential energy difference,

$U_{\mathrm{P}}-U_{\mathrm{R}}$

$\therefore \Delta U=U_{\mathrm{P}}-U_{\mathrm{R}}$

$\therefore \Delta U=W_{\mathrm{RP}}$

Therefore, we can define electric potential energy difference between two points as the work required to be done by an external force in moving (without accelerating) charge $q$ from one point to another for electric field of any arbitrary charge configuration.

Following comments may be made :

$(i)$ The right side of equation $(1)$ depends only on the initial and final positions of the charge.

- It means that the work done by an electrostatic field in moving a charge from one point to another depends only on the initial and the final points and is independent of the path taken to go from one point to the other. This is the fundamental characteristic of a conservative force.

$(ii)$ The actual value of potential energy is not significant it is only the difference of potential energy that is significant.

The potential energy difference,

$\mathrm{U}_{\mathrm{P}}=\mathrm{U}_{\mathrm{R}}$ $\therefore \Delta \mathrm{U} =\mathrm{U}_{\mathrm{P}}-\mathrm{U}_{\mathrm{R}}$ $\therefore \Delta \mathrm{U} =\mathrm{W}_{\mathrm{RP}}$

If potential energy is zero at infinity and adding an arbitrary constant $\alpha$ to potential energy at every point then,

$\left(\mathrm{U}_{\mathrm{P}}+\alpha\right)-\left(\mathrm{U}_{\mathrm{R}}+\alpha\right)=\mathrm{U}_{\mathrm{P}}-\mathrm{U}_{\mathrm{R}}$

$\therefore\left(\mathrm{U}_{\mathrm{P}}+\alpha-\mathrm{O}-\alpha\right)=\mathrm{U}_{\mathrm{P}}-\mathrm{U}_{\mathrm{R}}$

$\therefore\mathrm{U}_{\mathrm{P}}=\mathrm{U}_{\mathrm{P}}-\mathrm{U}_{\mathrm{R}}$

The work done for bringing a charge from point $\mathrm{R}$ to $\mathrm{P}$ at infinity distance,

$\mathrm{W}_{\mathrm{RP}}=\mathrm{U}_{\mathrm{P}} \text { or } \mathrm{W}_{\infty \mathrm{P}}=\mathrm{U}_{\mathrm{P}}$

Since above equation provides a definition of potential energy of charge at any point $P$. "Potential energy of charge $q$ at a point is the work done by the external force (equal and opposite to the electric field) in bringing the charge $q$ from infinity to that point".