Potential Energy of a Single Charge

Concept First

The potential energy of a single charge in an external field is different from the potential energy of a system of charges (Section 2.7).

• In Section 2.7, the field is produced by the charges themselves.
• Here, the field $\mathbf{E}$ (or potential $V$) is produced by other sources — not by the charge $q$ whose energy we are calculating.
• We assume $q$ is so small that it does not disturb the external sources. This is valid if $q$ is tiny, or if the external sources are held fixed by other forces.

The key question: How much work is done to bring a charge $q$ from infinity to a point in an already existing external field?

---

Work Done and Potential Energy

By definition, the electrostatic potential $V$ at a point $P$ is the work done in bringing a unit positive charge from infinity to $P$ (taking potential at infinity as zero).

Therefore, the work done in bringing a charge $q$ from infinity to $P$ is:

This work is stored as the potential energy of the charge $q$ in the external field.

If the point $P$ has position vector $\mathbf{r}$ relative to some origin, we write:

where:

• $U(\mathbf{r})$ = potential energy of charge $q$ at position $\mathbf{r}$,
• $V(\mathbf{r})$ = external electrostatic potential at $\mathbf{r}$ (due to sources other than $q$).

---

Important Clarification

• The external field $\mathbf{E}$ and potential $V$ may vary from point to point.
• We are not calculating the energy of the sources that produce the external field — only the energy of the charge $q$ placed in that field.

---

Example: Electron Volt (eV)

If an electron (charge $q = e = 1.6 \times 10^{-19} \, \text{C}$) is accelerated through a potential difference $\Delta V = 1 \, \text{V}$, the energy gained is:

This amount of energy is defined as 1 electron volt (1 eV): …