### What is Potential Energy?

Potential energy is an energy that is stored within an object, not in motion but capable of becoming active.

### Potential energy of a single charge in an electric field:

Let us consider a charge of magnitude q placed in an external electric field of magnitude E. Here the charge q under consideration is very small. The potential energy of the charge q in the field is equal to the work done in bringing the charge from infinity to the point. Here we note that, the external electric field E and the corresponding potential energy of the system vary from point to point in the field. Now, we know that the potential at infinity is always taken to be zero; the work done in bringing a charge from infinity to the point is given as qV.

The potential energy of the point q at a distance r from the origin in an external electric field is given as,

\(qV(r)\)

Where V(r) is the external potential at that point.

### Potential energy of a system of two charges in an electric field:

Let us consider a system of two charges q_{1} and q_{2 }located at a distance r_{1} and r_{2} from the origin. Let these charges be placed in an external field of magnitude E. Let the work done in bringing the charge q_{1} from infinity to r_{1} be given as q_{1}V(r_{1})and the work done in bringing the charge q_{2} from infinity to r_{2} against the external field can be given as q_{2}V(r_{2}). We note that, in the latter case, the work required to be done on q_{2} will include the field due to the charge q_{1} along with the electric field E, which can be given as,

\(\frac{q_{1}q_{2}}{4\pi \epsilon_{0} r_{12}}\)

Here, r_{12} is the distance between q_{1 }and q_{2}. By the 25, we can add these two to get the total work done in bringing q_{2} from infinity to r_{2}

\(q_{2}V(r_{2})+\frac{q_{1}q_{2}}{4\pi \epsilon_{0} r_{12}}\)

Thus, the total work done required to bring both the charges from infinity to the present configuration or the total potential energy of the system can be given as

\(q_{1}V(r_{1})+q_{2}V(r_{2})+\frac{q_{1}q_{2}}{4\pi \epsilon_{0} r_{12}}\)

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