In Section 2.7, the source of the electric field was specified – the charges and their locations - and the potential energy of the system of those charges was determined. In this section, we ask a related but a distinct question. What is the potential energy of a charge q in a given field? This question was, in fact, the starting point that led us to the notion of the electrostatic potential (Sections 2.1 and 2.2). But here we address this question again to clarify in what way it is different from the discussion in Section 2.7.
The main difference is that we are now concerned with the potential energy of a charge (or charges) in an external field. 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).Theexternal sources may be known, but often they are unknown or unspecified; what is specified is the electric field E or the electrostatic potential V due to the external sources. We assume that the charge q does not significantly affect the sources producing the external field. This is true if q is very small, or the external sources are held fixed by other unspecified forces. Even if q is finite, its influence on the external sources may still be ignored in the situation when very strong sources far away at infinity produce a finite field E in the region of interest. Note again that we are interested in determining the potential energy of a given charge q (and later, a system of charges) in the external field; we are not interested in the potential energy of the sources producing the 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. (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. 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
$=qV(\mathbf{r})$
Potential energy $V(\mathbf{r})$ of q at r in an external field $\mathbf{r}$.
Thus, if an electron with charge $q = e = 1.6 \times 10^{–19} \ C$ is accelerated by a potential difference of $\Delta V = 1 \ volt$, it would gain energy of $q \Delta V = 1.6 \times 10^{–19} \ J$. This unit of energy is defined as 1 electron volt or $1 \ eV$, i.e., $1 \ eV =1.6 \times 10^{–19} \ J$. The units based on $eV$ are most commonly used in atomic, nuclear and particle physics,
$(1 \ keV = 10^3 \ eV = 1.6 \times 10^{–16} \ J$,
$1 \ MeV = 10^6 \ eV = 1.6 \times 10^{–13} \ J$,
$1 \ GeV = 10^9 \ eV = 1.6 \times 10^{–10} \ J$ and
$1 \ TeV = 10^{12} \ e V = 1.6 \times 10^{–7}J$).
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