If another reference (ground, an equipotential surface, etc.) is used it must be stated explicitly and the potential is measured relative to that surface.
\[
V = \frac{W{\infty\to P}}{q{\text{test}}}\qquad\bigl[{\rm V}= {\rm J\,C^{-1}}\bigr],
\]
where \(W_{\infty\to P}\) is the external work required to move the test charge from infinity to point \(P\).
For a point charge \(Q\) the electric field is
\[
\mathbf{E}(r)=\frac{1}{4\pi\varepsilon_{0}}\frac{Q}{r^{2}}\hat{r}.
\]
The work to bring a unit test charge from \(\infty\) to a distance \(r\) is
\[
W{\infty\to r}= -\int{\infty}^{r}\mathbf{E}\!\cdot d\mathbf{s}
= -\int{\infty}^{r}\frac{1}{4\pi\varepsilon{0}}\frac{Q}{s^{2}}\,ds
= \frac{Q}{4\pi\varepsilon_{0}r}.
\]
Hence the electric potential due to a single point charge is
\[
\boxed{V(r)=\frac{Q}{4\pi\varepsilon_{0}r}=k\frac{Q}{r}}\qquad\bigl[{\rm V}= {\rm J\,C^{-1}}\bigr],
\]
with \(k=1/(4\pi\varepsilon_{0})\approx 8.99\times10^{9}\;{\rm N\,m^{2}\,C^{-2}}\).
Because electric potential is a scalar, the total potential at a point is the algebraic sum of the contributions from each charge:
\[
\boxed{V{\text{total}} = \sum{i}\frac{k\,Q{i}}{r{i}}}
\]
where \(r{i}\) is the distance from charge \(Q{i}\) to the point of interest.
Example (two‑charge system)
\[
V(P)=k\frac{Q{1}}{r{1}}+k\frac{Q{2}}{r{2}}.
\]
\[
\Delta V = V{B}-V{A}= -\int_{A}^{B}\mathbf{E}\!\cdot d\mathbf{s}.
\]
For a constant field \(\mathbf{E}=E\,\hat{x}\),
\[
V(x)= -Ex + C,
\]
so the potential falls linearly with distance. The constant \(C\) is set by the chosen reference (often \(V=0\) at \(x=0\)).
If a charge \(q\) is placed in the potential \(V\) created by another charge \(Q\), the electric potential energy of the pair is
\[
U = qV = \frac{k\,Qq}{r}= \frac{Qq}{4\pi\varepsilon_{0}r}.
\]
For \(n\) point charges the total electrostatic potential energy is
\[
\boxed{U = \frac{1}{2}\sum{i=1}^{n}\sum{\substack{j=1\\j\neq i}}^{n}\frac{k\,Q{i}Q{j}}{r_{ij}}}
= \frac{1}{2}\sum{i}\sum{j\neq i}\frac{Q{i}Q{j}}{4\pi\varepsilon{0}r{ij}}.
\]
The factor \(\tfrac12\) avoids double‑counting each pair \((i,j)\).
Two point charges \(Q=+5.0\;\mu\text{C}\) and \(q=-2.0\;\mu\text{C}\) are 0.10 m apart. Find \(U\).
\[
U=\frac{(8.99\times10^{9})(5.0\times10^{-6})(-2.0\times10^{-6})}{0.10}
=-0.090\;\text{J}.
\]
Charges: \(Q{1}=+3\;\mu\text{C}\) at the origin, \(Q{2}=+2\;\mu\text{C}\) at \((0,0.04\text{ m})\), \(Q_{3}=-1\;\mu\text{C}\) at \((0,0.06\text{ m})\). Find the total potential energy.
\(r{12}=0.04\;\text{m},\; r{13}=0.06\;\text{m},\; r_{23}=0.02\;\text{m}\).
\[
U=\frac{k}{2}\!\left(\frac{Q{1}Q{2}}{r{12}}+\frac{Q{1}Q{3}}{r{13}}+\frac{Q{2}Q{3}}{r_{23}}\right).
\]
\[
U\approx\frac{8.99\times10^{9}}{2}\Bigl[\frac{(3\times10^{-6})(2\times10^{-6})}{0.04}
+\frac{(3\times10^{-6})(-1\times10^{-6})}{0.06}
+\frac{(2\times10^{-6})(-1\times10^{-6})}{0.02}\Bigr]
\approx -0.045\;\text{J}.
\]
| Quantity | Symbol | Formula | Units |
|---|---|---|---|
| Electric potential (point charge) | V | \(V = \dfrac{kQ}{r}= \dfrac{Q}{4\pi\varepsilon_{0}r}\) | volt (V) |
| Potential of many point charges | V | \(V = \displaystyle\sum{i}\frac{kQ{i}}{r_{i}}\) | volt (V) |
| Potential difference | \(\Delta V\) | \(\Delta V = -\displaystyle\int_{A}^{B}\mathbf{E}\!\cdot d\mathbf{s}\) | volt (V) |
| Electric field from potential | \(\mathbf{E}\) | \(\mathbf{E}= -\nabla V\) (or \(E_{x}= -\dfrac{dV}{dx}\) in 1‑D) | newton per coulomb (N C⁻¹) |
| Potential energy (two charges) | U | \(U = \dfrac{kQq}{r}= \dfrac{Qq}{4\pi\varepsilon_{0}r}\) | joule (J) |
| Potential energy (many charges) | U | \(U = \dfrac{1}{2}\displaystyle\sum{i}\sum{j\neq i}\frac{kQ{i}Q{j}}{r_{ij}}\) | joule (J) |
| Permittivity of free space | \(\varepsilon_{0}\) | \(8.854\times10^{-12}\;\text{C}^{2}\text{N}^{-1}\text{m}^{-2}\) | farad per metre (F m⁻¹) |
In the laboratory you can measure a potential difference with a voltmeter. For a uniform field between parallel plates, the measured \(\Delta V\) between two points a distance \(d\) apart should satisfy \(\Delta V = Ed\). This provides a quick check of both the field strength and the calibration of the instrument.
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