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doPhysics · Electromagnetism 1

Conductors in Electrostatics

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Why this chapter
Why this chapter exists

Much of the charge we meet sits on metal: the dome of a Van de Graaff machine, a car body, the wires of a house. Metal has charge that is free to move, and that one fact decides everything in this chapter. The free charge moves until no field is left inside the metal. From that follow the charge on the surface, the field at sharp points, the calm inside a Faraday cage and the rules for earthing. Gauss's law, from the previous chapter, does most of the work.

You will learn
By the end of this chapter you will
  1. Explain why the field inside a conductor is zero, and deduce that its charge sits on the surface and that it is an equipotential.
  2. Use \(E = \sigma/\varepsilon_0\) just outside a conductor, and explain why a sharp point carries the densest charge.
  3. Find the charges on the walls of a cavity and on the outer surface, and explain electrostatic shielding.
  4. Find the charges and potentials of concentric shells, and say what earthing and joining conductors fix.
  5. Find the electrostatic pressure on a charged surface and the largest charge a sphere can hold in air.
Before you begin
Before you begin

From the chapter Electric Flux and Gauss's Law you should know Gauss's law. You should also know the fields of a sheet and a thin shell, \(\sigma/2\varepsilon_0\) and \(kQ/r^2\). The chapter Electric Potential, Energy and the Dipole gives the potential as work per unit charge, and \(kq/r\) for a point charge. The chapter on Surface Tension gives the excess pressure of a soap bubble, used once in Lesson 5.

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Lessons

1
What a Conductor Does in a Field
The free electrons of a metal move until the field inside it is zero. So any extra charge sits on the surface, the whole body is at one potential, and the field just outside is \(\sigma/\varepsilon_0\) along the normal.
Free electrons and zero field insideCharge lives on the surfaceThe conductor is an equipotentialField just outside a conductor
14 min0%
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2
Charge on the Surface and the Sharp End
A charged metal sphere has no field inside and one potential kQ/R throughout. Joined bodies share one potential, so charge crowds where the surface curves most sharply. That is why a lightning conductor ends in a point.
Field and potential of a charged metal sphereCharge crowds to the sharp endSpheres at the same potential
13 min0%
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3
Cavities, Shielding and the Faraday Cage
A charge in the cavity of a metal shell draws an equal and opposite charge onto the cavity wall, and the outside sees only the total. An empty cavity has no field, whatever lies outside. That is shielding, and it explains the lift and the car.
A charge inside a cavityMoving the charge inside the cavityElectrostatic shieldingA hollow conductor in a uniform field
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4
Concentric Shells and Earthing
The charges on nested metal shells follow from a Gaussian surface in each shell's metal, and their potentials add shell by shell. Earthing fixes a potential at zero and lets the charge follow; joining conductors gives them one potential.
Charges on concentric shellsPotentials of concentric shellsEarthing a shellJoining shells and spheres
16 min0%
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5
Electrostatic Pressure and the Limits of Air
Every patch of a charged conductor is pushed outward with a pressure \(\sigma^2/2\varepsilon_0 = \tfrac12 \varepsilon_0 E^2\). The push can hold a charged soap bubble open. Air breaks down near \(3 \times 10^6\,\text{V/m}\), which caps the charge a sphere can hold and makes sharp points leak.
Force on a patch of a charged surfaceElectrostatic pressure in numbersThe charged soap bubbleBreakdown of air and the largest charge
14 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

cavity
A hollow space inside a conductor, completely surrounded by its metal.
concentric shells
Spherical shells that share one centre, one inside another.
conductor
A material with charge that is free to move through it, such as a metal with its free electrons.
corona discharge
A slow leak of charge into the air from a sharp point, where the field is strong enough to make the nearby air conduct. It shows as a faint glow with a hiss.
dielectric strength
The largest field an insulator can stand before it breaks down and conducts. For dry air it is about \(3 \times 10^6\,\text{V/m}\).
earthing
Joining a conductor to the Earth by a wire, which fixes its potential at zero. Also called grounding.
electrostatic equilibrium
The state of a conductor when all its charges are at rest. The field inside its material is then zero.
electrostatic pressure
The outward force per unit area on the surface of a charged conductor, \(\sigma^2/2\varepsilon_0 = \tfrac12\varepsilon_0E^2\).
electrostatic shielding
The absence of any electric field inside an empty cavity of a conductor, whatever charges and fields lie outside it.
equipotential
A surface or a body every point of which is at the same potential. A conductor in electrostatic equilibrium is one.
Faraday cage
A closed box or net of metal used to keep electric fields out of the space inside it.
free electrons
The outer electrons of the atoms of a metal, not bound to any one atom and free to move through the whole body.
induced charge
Charge that gathers on parts of a conductor because of a nearby field or charge, while the conductor's total charge stays the same.
lightning conductor
A pointed metal rod above the top of a building, joined by a thick strip to a plate buried in the earth. It gives lightning a safe path to the ground.
surface charge density
The charge on each square metre of a surface, \(\sigma\), in C/m². On a conductor it can vary from place to place.
Van de Graaff generator
A machine that carries charge on a moving belt to the inside of a large metal dome. It builds up a high potential on the dome.