Potential energy and conductors

Potential energy of two charges: the work needed to bring them from far apart to a distance r:

U = k q₁q₂ / r

Keep the signs. Positive U (like charges): work had to be done to push them together. Negative U (unlike charges): they pulled themselves together, and work is needed to separate them.

Example: two 1 µC charges, 10 cm apart: U = 9 × 10⁹ × 10⁻⁶ × 10⁻⁶ / 0.1 = 0.09 J.

For three or more charges, add U for every pair: q₁q₂, q₂q₃, q₁q₃. Three charges have three pairs; four charges have six.

In an external field, a charge q at potential V has energy qV, in addition to the pair energies.

Dipole in a uniform field: U = –pE cos θ. It is lowest (–pE) when p lines up with E, which is the stable position, and highest (+pE) when it points the opposite way. Turning a dipole from θ₁ to θ₂ needs work pE(cos θ₁ – cos θ₂).

Conductors in electrostatics behave in a fixed way, because their free electrons move until everything settles:

• E = 0 inside the conductor.
• Any extra charge sits on the outer surface.
• The whole conductor is at one potential (it is an equipotential).
• Just outside, E is perpendicular to the surface and equals σ/ε₀.
• Charge gathers most at sharp points, where σ is largest. This is how lightning conductors work.