Case 1 : Potential energy of a single charge in presence of external electric field :
The external electric field E and the corresponding external potential V may vary from point to point. By definition, V at a point P is the work done in bringing a unit positive charge from infinity to the point P. Thus, work done in bringing a charge q from infinity to the point P in the external field is qV. This work is stored in the form of potential energy of q. If the point P has position vector r relative to some origin, we can write: Potential energy of q at r in an external field
= qV(r) --------- (1)
where V(r) is the external potential at the point r.
Thus, if an electron with charge q = e = 1.6×10–19 C is accelerated by a potential difference of ΔV = 1 volt, it would gain energy of qΔV = 1.6 × 10–19J. This unit of energy is defined as 1 electron volt or 1eV, i.e., 1 eV = 1.6 × 10–19J.
Case 2 : Potential energy of a system of two point charges in presence of external electric field :
The potential energy of a system of two charges q1 and q2 located at vector r1 and r2, respectively, in an external field can be calculated by knowing the work done in bringing the charge q1 from infinity to vector r1.
Work done in this step is q1 V(vector r1), using Eq. (1). Next, we consider the work done in bringing q2 to vector r2. In this step, work is done not only against the external field vector E but also against the field due to q1.
Work done on q2 against the external field = q2 V (r2)
Work done on q2 against the field due to q1
where r12 is the distance between q1 and q2. We have made use of equation. By the superposition principle for fields, we add up the work done on q2 against the two fields (vector E and that due to q1):
Work done in bringing q2 to r2
Thus, Potential energy of the system = the total work done in assembling the configuration
