# Potential energy of charges in an electric field

### 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 q1 and q2 located at a distance r1 and r2 from the origin. Let these charges be placed in an external field of magnitude E.  Let the work done in bringing the charge q1 from infinity to r1 be given as q1V(r1)and the work done in bringing the charge q2 from infinity to r2­ against the external field can be given as q2V(r2). We note that, in the latter case, the work required to be done on q2 will include the field due to the charge q1 along with the electric field E, which can be given as,

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

Here, r12 is the distance between q1 and q2. By the 25, we can add these two to get the total work done in bringing q2 from infinity to r2

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

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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