AcademyElectric Potential
Academy
Calculating Potential
Level 1 - Physics topic page in Electric Potential.
Principle
Electric potential from multiple sources is found by scalar superposition, adding signed contributions rather than vector components.
Notation
\(V\)
electric potential at a field point
\(\mathrm{V}\)
\(Q_i\)
source point charge
\(\mathrm{C}\)
\(R_i\)
distance from source \(i\) to the field point
\(\mathrm{m}\)
\(dq\)
small source charge element
\(\mathrm{C}\)
\(\lambda\)
linear charge density
\(\mathrm{C\,m^{-1}}\)
\(dl\)
small length element
\(\mathrm{m}\)
\(k\)
Coulomb constant
\(\mathrm{N\,m^{2}\,C^{-2}}\)
Method
Derivation 1: Add point-charge potentials
Potential is scalar, so each point charge contributes a signed number at the field point. There are no \(x\)- and \(y\)-components to resolve.
One source
\[V_i=k\frac{Q_i}{R_i}\]
Scalar superposition
\[V=\sum_i V_i\]
Point-charge sum
\[V=k\sum_i\frac{Q_i}{R_i}\]
Derivation 2: Continuous charge
For a continuous distribution, split the charge into elements. Each element contributes a small scalar potential.
Small contribution
\[dV=k\frac{dq}{R}\]
Line element
\[dq=\lambda\,dl\]
Integrate
\[V=\int k\frac{dq}{R}\]
Derivation 3: Use symmetry without directions
When distances are equal, potential contributions combine by charge sign alone.
Equal distances
\[R_1=R_2=R\]
Two-charge potential
\[V=k\frac{Q_1+Q_2}{R}\]
Equal opposite charges
\[Q_1=-Q_2\Rightarrow V=0\]
The electric field at the same point may still be nonzero.
Rules
These are the compact potential-calculation tools.
Point source
\[V_i=k\frac{Q_i}{R_i}\]
Scalar superposition
\[V=\sum_i V_i=k\sum_i\frac{Q_i}{R_i}\]
Continuous source
\[dV=k\frac{dq}{R},\qquad V=\int k\frac{dq}{R}\]
Line charge
\[dq=\lambda\,dl\]
Examples
Question
Charges
\[+4.0\,\mathrm{nC}\]
and \[-1.0\,\mathrm{nC}\]
are each \[0.50\,\mathrm{m}\]
from point P. Find \(V_P\).Answer
\[V_P=k\frac{Q_1+Q_2}{R}=\frac{(8.99\times10^9)(3.0\times10^{-9})}{0.50}=54\,\mathrm{V}\]
Checks
- Add potential contributions with signs, not directions.
- A zero potential point can still have nonzero electric field.
- Distances \(R_i\) are always from source charge to the field point.
- Continuous-source integrals need the correct charge element \(dq\).