Equipotential Surfaces

If you trace out all points at the same potential, you get an equipotential surface — a contour map of $V$. These surfaces reveal the geometry of the field at a glance.

Concept

An equipotential surface is a surface on which $V$ has the same value at every point.

Properties:

1. No work is done in moving a charge along an equipotential surface, because $W = q\,\Delta V = 0$.
2. The electric field is perpendicular to the equipotential surface at every point. If there were a tangential component, moving along the surface would do work — contradiction.
3. Equipotential surfaces are closer together where the field is stronger (just like contour lines are closer on steep terrain).
4. Two different equipotential surfaces never intersect — a point would then have two potentials, impossible.
5. Surface of a conductor in electrostatic equilibrium is itself an equipotential.

Relation $\vec E = -\nabla V$. In one dimension: $E = -\frac{dV}{dx}.$ In three dimensions: $\vec E = -\hat x\frac{\partial V}{\partial x} - \hat y\frac{\partial V}{\partial y} - \hat z\frac{\partial V}{\partial z}.$ This says the field points in the direction of steepest descent of $V$, with magnitude equal to the slope.

Derivation

For a small displacement $d\vec l$ along an equipotential, $dV = 0$. But $dV = -\vec E\cdot d\vec l.$ So $\vec E\cdot d\vec l = 0$ for every direction tangent to the surface — meaning $\vec E$ is perpendicular to the surface.

From $V_B - V_A = -\int_A^B \vec E\cdot d\vec l$, take infinitesimal $d\vec l$: $dV = -\vec E\cdot d\vec l = -E_l\,dl,$ where $E_l$ is the component of $\vec E$ along $d\vec l$. Thus $E_l = -dV/dl$, giving the gradient relation.

Examples of Equipotential Surfaces

• Point charge: concentric spheres around the charge.
• Uniform field (e.g. between parallel plates): parallel planes perpendicular to the field.
• Electric dipole: complex non-spherical surfaces; the equatorial plane is a single equipotential at $V = 0$.
• Conductor: the entire conductor (interior + surface) is one equipotential.

Worked Example

A uniform field $E = 200\,V/m$ points along $+\hat x$. Sketch the equipotentials.

Since $E_x = -dV/dx = 200$, we get $V = -200x + C$. Equipotentials are planes $x =$ const. Moving by $1\,cm$ along $\hat x$ changes $V$ by $-2\,V$.

Conversely, given $V(x,y,z) = 5x^2 + 3y$, the field is $\vec E = -\nabla V = -10x\,\hat x - 3\,\hat y.$ At $(1,0,0)$, $E = -10\hat x - 3\hat y$, magnitude $\sqrt{100+9} = \sqrt{109} \approx 10.4\,V/m$.

Common Confusions

• Equipotential is NOT the same as a field line. Equipotentials are perpendicular to field lines.
• Conductor surface is an equipotential. Interior is too (since $E_\text{in} = 0$).
• $V$ constant does not imply $E = 0$ everywhere — only the tangential component to that surface. But if $V$ is constant in a region, then $E = 0$ in that region.
• Equipotentials cannot intersect.

Key Takeaways

• Equipotential = surface of constant $V$.
• $\vec E \perp$ equipotential surfaces, pointing toward lower $V$.
• $\vec E = -\nabla V$; magnitude is the rate of decrease of $V$.
• No work to move a charge on an equipotential.
• A conductor in equilibrium is an equipotential (interior and surface).