Potential due to a Point Charge

The single most important result of electrostatics is also the simplest: the potential of a point charge.

Concept

A point charge $q$ at the origin produces, at a distance $r$, the potential $V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r} = \frac{kq}{r}.$ The reference $V(\infty) = 0$ is used. The sign of $V$ matches the sign of $q$: a positive charge produces positive potential (high places need work to reach with a positive test charge); a negative charge produces negative potential.

Two key consequences:

• Equipotential surfaces of a point charge are concentric spheres.
• Superposition: for many charges, $V_\text{net} = \sum_i kq_i/r_i$. No vectors needed.

Derivation

Start from $V(P) = -\int_\infty^P \vec E\cdot d\vec l$. The field is radial: $\vec E = \dfrac{kq}{r^2}\hat r$. Choose a radial path so $d\vec l = dr\,\hat r$: $V(r) = -\int_\infty^r \frac{kq}{r'^2}dr' = kq\left[\frac{1}{r'}\right]_\infty^r = \frac{kq}{r}.$

Notice the integral converges thanks to the $1/r^2$ fall-off of $E$. For infinite line or sheet distributions, $V(\infty)$ is not zero — we must pick a different reference.

Worked Example

Find the potential at the centre of a square of side $1\,m$ with charges $+1, +2, +3, +4\,\mu C$ at the corners.

Distance from centre to each corner: $r = \sqrt{0.5^2 + 0.5^2} = \tfrac{1}{\sqrt 2}\,m \approx 0.707\,m$.

$V = k\sum_i \frac{q_i}{r} = \frac{9\times 10^9}{0.707}\,(1+2+3+4)\times 10^{-6} = 1.273\times 10^{10}\times 10^{-5} = 1.273\times 10^5\,V.$

Now suppose two of the corner charges are $-1, -2\,\mu C$ instead. The same formula applies, but now $\sum q_i = +1-1+3-2 = +1\,\mu C$. So $V = \frac{9\times 10^9 \times 1\times 10^{-6}}{0.707} = 1.27\times 10^4\,V.$

Common Confusions

• $V$ falls as $1/r$ — not $1/r^2$. That is the field, not the potential.
• Potential is scalar. Even for many charges, you add numbers, not vectors.
• Sign of $V$ matches sign of $q$. Be careful with negative charges.
• Potential at the location of a point charge is infinite. This is an idealization; real charge distributions give finite values.
• Reference at infinity works only for localized charges. For a long wire or infinite sheet, you must pick a finite reference.

Key Takeaways

• $V(r) = kq/r$, with sign tracking $q$.
• Equipotentials are concentric spheres for a point charge.
• Net potential from many charges: $V = k\sum q_i/r_i$ (algebraic scalar sum).
• $V \propto 1/r$, slower decay than the field's $1/r^2$.
• Reference $V(\infty) = 0$ requires a localized source.