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28.4 Electrostatics and Potentials

As we have noted several times by now, Coulomb's law states that like electric charges repel and unlike ones attract with an inverse square law of force. When attracting this force is very like that of gravity but acting between charged particles proportionally to each of their charges rather than to their masses. The statements below all refer to electrostatic fields: fields produced by charge densities that are not changing in time; not to electric fields produced by varying charge densities.

Here now are some mathematical implications of this law:

1. We can express the electric field of a charged particle with charge e at point P by the formula. (This is Coulomb's law.)

E ⟶ ( P ) = e u ^ P − P ' | P − P ' | 2

2. The divergence of this field is 0 except when P − P ' . (By differentiation.)

3. The flux of this field over a sphere around P ' is 4 π e . (By integration.)

4. The flux of this field over the boundary of any volume containing P ' is 4 π e , and over any other volume is 0. (By the divergence theorem.)

5. The field of a sum of charges is the sum of the fields of each one. (Physical fact.)

6. The integral of the density of a point charge over any volume containing it is e , and is 0 if the volume does not contain the charge. (Definition of density.)

7. The density of a sum of charges is the sum of their densities.

8. For a point charge at P ' the flux of E ⟶ around the boundary of V is 4 π times the integral of its charge density over V . (By 4, 5, 6, 7.)

9. The integral over V of ∇ ⟶ · E ⟶ − 4 π ρ is 0. (By the divergence theorem.)

10. The curl of E ⟶ is 0 except at P = P ' for a point charge at P ' . ( E ⟶ points outward and depends only on distance in the outward direction = by differentiation.)

11. The circulation of the electric field E ⟶ of a point charge is 0 over any closed path. (By Stokes theorem and 10.)

12. The same holds for any arrangement of charges.

13. For any charge distribution, E ⟶ ( P ) can be written as − ∇ ⟶ V for a "potential function" V ( P ) . (Define V as the line integral of − E ⟶ from some point to P . This is uniquely defined by 11.)

14. V ( P ) satisfies the equation

( ∇ ⟶ · ∇ ⟶ ) V = − 4 π ρ

15. V P ' ( P ) for a point charge at P ' is e | P − P ' | and it obeys the equation

( ∇ ⟶ · ∇ ⟶ ) V P ' = − e 4 π δ ( 3 ) ( P − P ' )

(Check that − ∇  V P ' is the electric field of a point charge at P '. )

16. The potential determined by a charge density ρ ( P ) in free space (ignoring any matter in it) is given by

V ( P ) = ∭ ρ ( P ' ) | P − P ' | d τ ' (By integration using 15.)

17. A conductor is a material on which electric charge is free to move. In a static situation, the potential on a conductor must be constant.

18. A plane conductor acts like a mirror, so that the potential from a charge density ρ ( P ) bounded by a plane conductor is

V ( P ) = ∭ ρ ( P ' ) ( 1 | P − P ' | − 1 | P − P " | ) d τ '

where P " is the reflection of the point P ' in the plane.

Exercises:

28.1 Go over each statement above. Prove it to your satisfaction. Explain it to some victim.

28.2 Suppose you have a charge distribution ρ ( P ) and it is bounded on two sides by planar conductors at right angles. What potential function would it produce? (Hint: try it with two mirrors.)

28.3 Suppose your distribution was bounded by conductors in three perpendicular planes. What potential function would you find?

Summary:

The electrostatic field of a fixed charge distribution ρ obeys the equations

∇ ⟶ · E ⟶ = 4 π ρ ∇ ⟶ · E ⟶ = 0

It can be written as the gradient of a scalar field, − V called a potential, which therefore obeys the equations

E ⟶ = − ∇ ⟶ V ( ∇ ⟶ · ∇ ⟶ ) V = − 4 π ρ

The potential at P of a point charge at P ' is

e | P − P ' |

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