AcademyElectric Potential
Academy
Equipotential Surfaces
Level 1 - Physics topic page in Electric Potential.
Principle
An equipotential surface is a set of points with the same electric potential, so moving along it requires no electric work.
Notation
\(V\)
electric potential
\(\mathrm{V}\)
\(\Delta V\)
potential difference between two points
\(\mathrm{V}\)
\(q\)
charge moved between points
\(\mathrm{C}\)
\(W_{\mathrm{elec}}\)
work done by the electric force
\(\mathrm{J}\)
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(d\vec \ell\)
small displacement along a path
\(\mathrm{m}\)
Method
Derivation 1: Work along an equipotential
If two points have the same potential, a charge moving between them has no potential-energy change.
Equipotential condition
\[\Delta V=0\]
Energy change
\[\Delta U=q\Delta V=0\]
Electric work
\[W_{\mathrm{elec}}=-q\Delta V=0\]
Derivation 2: Why field lines meet equipotentials at right angles
For a tiny displacement along an equipotential, the electric work must be zero. Since \(W=q\vec E\cdot d\vec\ell\), the field has no component tangent to the surface.
Small electric work
\[dW=q\vec E\cdot d\vec\ell\]
Along equipotential
\[dW=0\]
No tangent component
\[\vec E\cdot d\vec\ell=0\]
Perpendicular crossing
\[\vec E\perp\text{equipotential surface}\]
Derivation 3: Conductors in electrostatic equilibrium
Inside conducting material in electrostatic equilibrium, \(\vec E=0\). If there is no field along the conductor, there is no potential change through it.
Conductor equilibrium
\[\vec E=0\quad\text{inside conducting material}\]
No potential change
\[\Delta V=0\]
Conductor surface
\[V=\mathrm{constant}\]
Rules
These are the compact equipotential rules.
Equipotential
\[\Delta V=0\]
No electric work
\[W_{\mathrm{elec}}=-q\Delta V=0\]
Perpendicular field
\[\vec E\perp\text{equipotential surface}\]
Conductor surface
\[V=\mathrm{constant}\quad\text{on a conductor in electrostatic equilibrium}\]
Examples
Question
A
\[+2.0\,\mathrm{nC}\]
charge moves along a path where \[V=40\,\mathrm{V}\]
everywhere. Find the electric work.Answer
The path is equipotential, so
\[\Delta V=0\]
\[W_{\mathrm{elec}}=-q\Delta V=0\]
Checks
- Equipotential does not mean zero potential; it means constant potential.
- Moving along an equipotential gives zero electric work for any charge.
- Electric field lines cross equipotentials at right angles.
- Closely spaced equipotentials usually indicate a stronger field.