Field and potential, equipotential surfaces, and conductors

Derivation: relation between E and V

A and B are two close points in a uniform field E, a small distance Δr apart, with B farther along the field.

1. 1
   Move a unit positive charge from A to B. The field pushes it along the motion, so the outside agent must push against the motion.
2. 2
   Work done by the agent on the unit charge = EΔr cos 180° = −EΔr. This is the potential difference ΔV = V(B) − V(A).
3. 3★
   So, E = −ΔV/Δr. The field is the negative potential gradient: it points the way the potential falls fastest.
4. 4
   Between two points a distance d apart along a uniform field: E = (V_A − V_B)/d.
5. 5
   Hence the unit of E can also be written V m⁻¹, the same as N C⁻¹. Where V is the same everywhere, E = 0.

★ key step

Equipotential surfaces

An equipotential surface is a surface on which every point has the same potential. For a point charge, V = kq/r depends only on r, so its equipotentials are concentric spheres. In a uniform field they are parallel planes at right angles to the field. For a dipole, the whole equatorial plane is the V = 0 surface. Moving a charge along an equipotential needs no work, because W = q × ΔV = 0. So the field has no part along the surface: E is always at right angles to an equipotential surface.

Properties of equipotential surfaces

• No work is done in moving a charge along an equipotential surface.
• The electric field is always perpendicular to the surface at every point.
• Two equipotential surfaces never intersect. At the crossing, one point would have two potentials.
• For equal steps of potential, the surfaces are closer together where the field is strong.
• The surface of a conductor in electrostatics is an equipotential surface.

How to draw equipotential surfaces

1. 1
   Point charge: dashed concentric circles round +q. Draw solid radial field lines crossing them at right angles. For equal steps of V, the circles get farther apart going outward.
2. 2
   Uniform field: solid horizontal field lines with arrows; dashed vertical straight lines, equally spaced, crossing them at right angles.
3. 3
   Dipole: dashed closed curves round each charge, crowded near the charges. The perpendicular bisector is the V = 0 surface. Field lines cross every dashed curve at right angles.
4. 4
   Two equal positive charges: dashed closed curves round each charge. Farther out they merge into one curve shaped like a peanut, and far away they look like circles.

Conductors in an electric field

• A conductor, such as a metal, has free electrons that move easily. An insulator, such as wood, glass, mica or ebonite, has none.
• In an outside field E, the free electrons drift against E and pile up on one face. The opposite face is left positive.
• These surface charges make a field opposite to E. The electrons stop when the two fields cancel, in about 10⁻¹⁶ s.
• So, in electrostatic equilibrium, E = 0 inside a conductor, and the whole conductor is at one potential.
• Any charge on a conductor sits on its outer surface, and the field just outside is at right angles to the surface.
• Electrostatic shielding: the field inside an empty cavity in a conductor is zero, whatever the field outside. So a car or bus is safer than open ground in a lightning storm, and delicate instruments are kept in metal boxes.

Worked example: field from potentials