Equipotential Surfaces

Definition

• An equipotential surface is a surface at every point of which the electric potential is the same.
• For a single point charge, $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$, so $V$ is constant wherever $r$ is constant: the equipotential surfaces are concentric spheres centred on the charge. For a line charge, the surfaces are coaxial cylinders.
• Equipotential surfaces can be drawn through any region in which there is an electric field.

No work is done on an equipotential surface

• By definition, the potential difference between two points P and Q is the work done per unit positive charge in displacing it from Q to P: $V_P - V_Q = W_{QP}$.
• If P and Q lie on the same equipotential surface then $V_P = V_Q$, hence

$$\boxed{\,W_{QP} = 0\,}$$

• No work is required to move a test charge along an equipotential surface.

The electric field is normal to an equipotential surface

First argument, from the work

• Let $dx$ be a small displacement over the equipotential surface through which unit positive charge is carried. Then

$$dW = \vec{E}\cdot d\vec{x} = E\,dx\cos\theta = 0$$

• Since $E$ and $dx$ are not zero, $\cos\theta = 0$, i.e. $\theta = 90^\circ$. Hence $\vec{E}\perp d\vec{x}$: the electric field intensity is always normal to the equipotential surface. For any charge distribution, the equipotential surface through a point is normal to the electric field at that point.

Second argument, by contradiction

• If the field were not normal it would have a non-zero component along the surface. Moving a test charge against that component would need work to be done. But there is no potential difference between any two points of an equipotential surface, so no work can be required. Therefore the electrostatic field must be normal to the equipotential surface at every point, and conversely.

Equipotential surfaces never intersect

• If two equipotential surfaces intersected, the point of intersection would have two directions of the electric field (one normal to each surface), which is not possible.

Standard pictures to describe

• Isolated point charge: concentric spheres, drawn as concentric circles in a plane, closer together near the charge.
• Uniform electric field: a set of equally spaced parallel planes perpendicular to the field lines.
• Electric dipole: closed surfaces surrounding each charge, symmetric about the perpendicular bisector; the equatorial plane itself is the zero-potential surface.
• Two identical positive charges: separate closed surfaces around each charge at short range, merging into a single surface enclosing both at large distances.
• Between two plane metallic sheets: planes parallel to the sheets. When one sheet is replaced by a charged metallic sphere, the surfaces bend to follow the electrode shapes, spherical near the sphere and planar near the sheet.

What they are for

• Like the lines of force, equipotential surfaces give a visual picture of both the direction of the electric field (always normal to them) and its magnitude (the more crowded the surfaces, the stronger the field).

The standard numerical

• A charged particle held in equilibrium between two horizontal plates a distance $dx$ apart with a potential difference $dV$: get the field from

$$E = -\dfrac{dV}{dx}$$

• and then equate the gravitational and electrostatic forces, $mg = qE$, to find $m = \dfrac{qE}{g}$. Take $g = 9.8\ \text{m s}^{-2}$ and keep the signs of $q$ and $E$ together so that the mass comes out positive.