1. Concept Overview
In an electric field, different points have different electric potentials. An equipotential surface is a surface where every point has the same potential. This means that moving a charge anywhere along this surface requires no work. Equipotential surfaces help visualise electric potential in a simple, geometric way.
2. Definition
Equipotential surfaces are surfaces on which the electric potential is the same everywhere. Therefore, the potential difference between any two points on such a surface is zero.
3. Key Properties of Equipotential Surfaces
These surfaces follow certain rules that reflect how electric potential behaves.
3.1. No Work Done on the Surface
Since potential difference between any two points on the surface is zero:
\( W = q (V_B - V_A) = 0 \)
No work is needed to move a charge along an equipotential surface.
3.2. Equipotential Surfaces are Perpendicular to Electric Field Lines
The electric field is always directed perpendicular to equipotential surfaces. If it were not, work would be done when moving along the surface.
3.3. Closer Surfaces Mean Stronger Field
The spacing between equipotential surfaces indicates the field strength:
- Narrow spacing → strong electric field
- Wide spacing → weak electric field
4. Examples of Equipotential Surfaces
Different charge distributions create different geometric shapes of equipotential surfaces.
4.1. Point Charge
Equipotential surfaces are concentric spheres centred on the charge. As distance increases, the potential decreases but the spherical shape remains the same.
4.2. Electric Dipole
The surfaces are more complex, bending between the positive and negative charges. They mirror the dipole’s curved field pattern.
4.3. Uniform Electric Field
Equipotential surfaces are equally spaced parallel planes perpendicular to the field lines.
5. Relation Between Equipotentials and Potential Gradient
The electric field is linked to how fast potential changes in space.
5.1. Formula
The magnitude of the electric field is given by the gradient of potential:
\( E = -\dfrac{dV}{dr} \)
This means the electric field points in the direction of decreasing potential.
6. Movement Between Equipotential Surfaces
Moving a charge from one equipotential surface to another requires work. The amount of work depends on the potential difference between the two surfaces.
6.1. Work Done
\( W = q (V_B - V_A) \)
This work is positive or negative depending on whether the charge moves along or against the field.
7. Physical Interpretation
Equipotential surfaces act like “contour lines” of electric potential. They show where a charge would have the same potential energy. Just as a ball rolling downhill moves perpendicular to contour lines on a map, a charge moves perpendicular to equipotential surfaces—along electric field lines. This helps visualise the electric field and potential together.