Potential energy of a single charge :
The external field E is not produced by the given charge(s) whose potential energy we wish to calculate. E is produced by sources external to the given charge(s).
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. (We continue to take potential at infinity to be zero.) Thus, work done in bringing a charge q from infinity to the point P in the external field is qV.
Fig. Electric potential energy
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
U= q V(r)
where V(r) is the external potential at the point r.
Note :
Thus, if an electron with charge q = e = 1.6×10^–19 C is accelerated by a potential difference of ∆V = 1 volt, it would gain energy of q∆V = 1.6 × 10^–19J. This unit of energy is defined as 1 electron volt or 1eV, i.e., 1 eV=1.6 × 10^–19J. The units based on eV are most commonly used in atomic, nuclear and particle physics, (1 keV = 10^3 eV = 1.6 × 10^–16J, 1 MeV = 10^6eV = 1.6 × 10^–13J, 1 GeV = 10^9eV = 1.6 × 10^–10J and 1 TeV = 10^12eV = 1.6 × 10^–7J).
Potential energy of a system of charges in an external field :
For calculate potential energy of a system of two charges q1 and q2 located at r1 and r2, respectively, in an external field First, we calculate the work done in bringing the charge q1 from infinity to r1. Work done in this step is W1=q1 V(r1).
Next, we consider the work done in bringing q2 to r2. In this step, work is done not only against the external field E but also against the field due to q1.
Work done on q2 against the external field
W2 = q2 V (r2)
Work done on q2 against the field due to q1
W3 = q1q2 / 4πε0 r12 where r12 is the distance between q1 and q2.
By the superposition principle for fields, we add up the work done on q2 against the two fields (E and that due to q1): Work done in bringing q2 to r2
W23 = W2+W3
W23 = q2 V (r2) + q1q2 / 4π ε0 r12
Thus, Potential energy of the system = the total work done in assembling the configuration
U = W (= W1+W23)
U = q1 V(r1) + q2 V (r2) + q1q2 / 4π ε0 r12
