Electrostatics of Conductors

A conductor contains free charges. In electrostatic equilibrium, these charges redistribute themselves until the internal electric field becomes zero.

This redistribution leads to a few standard results:

1. The electric field inside a conductor is zero.
2. Excess charge resides on the outer surface.
3. Electric field at the surface is perpendicular to the surface.
4. Potential is the same throughout the conductor and on its surface.
5. A cavity inside a conductor remains shielded from external electrostatic influence if no charge is placed inside it.

The phenomenon in which the electric field inside a cavity of a conductor is zero, irrespective of external charges or fields, is called electrostatic shielding.

The condition in which charges in a conductor are at rest, and no further motion of charges occurs.

Surface charge density is the charge per unit area on the surface of a conductor and is denoted by \[\sigma\].

A conductor in electrostatic equilibrium is an equipotential body, meaning all points on it are at the same electric potential.

\[\vec{E}=\frac{\sigma}{\varepsilon_0}\hat{n}\]

where

• σ = surface charge density
• \[\hat n\] = outward normal unit vector
• \[\varepsilon_0\] = permittivity of free space.

Magnitude form:

E = \[\frac{\sigma}{\varepsilon_0}\]

Vector form:

$$\vec{E} = \frac{\sigma}{\varepsilon_0}\hat{n}$$

The electric field inside a conductor is zero

• If an electric field existed inside the conductor, free electrons would continue to move. Since equilibrium means no net motion of free charges, the electric field inside must be zero.
• Important Point: This statement is true only in electrostatic equilibrium.

Excess charge lies only on the outer surface

• If excess charge remained inside the conductor, it would create an internal electric field and disturb the equilibrium. Therefore, the excess charge stays on the outer surface.

The electric field at the surface is normal to the surface

• If the field had a tangential component, charges on the surface would move along the surface. Hence, at equilibrium, the electric field is perpendicular to the surface.

Potential is constant throughout the conductor

• Since the electric field inside the conductor is zero, no work is done in moving a test charge from one point to another inside it. Therefore, the entire conductor is at the same potential.

An empty cavity inside a conductor has zero field

• If no charge is placed inside the cavity, the electric field in the cavity remains zero. This is the basis of electrostatic shielding.

Using a small Gaussian pillbox at the surface of a charged conductor and applying Gauss's law, the flux comes only from the outer face because the field inside the conductor is zero. This gives the result \[E = \dfrac{\sigma}{\varepsilon_0}\].

For short answers: Mention “Gauss's law + zero field inside conductor” to justify the formula.

(a) Comb and bits of paper

• When a dry comb is run through dry hair, friction between hair and comb transfers electrons, so the comb becomes charged (usually negatively).
• A neutral paper piece has positive and negative charges bound in its molecules; in the presence of the charged comb, these charges rearrange slightly (polarisation), so one side of the paper is nearer and oppositely influenced, leading to net attraction.
• If hair is wet or it is a rainy day, moisture makes the surface slightly conductive, so charges leak away quickly and the comb does not retain enough charge, making attraction weak or absent.

(b) Slightly conducting aircraft tyres

• Ordinary rubber is a good insulator, so if tyres were perfectly insulating, friction with the runway and air would allow static charge to build up on the aircraft body and tyres.
• Special aircraft tyres are made slightly conducting so that any charge produced by friction can slowly leak to the ground, preventing a dangerous spark discharge that could ignite fuel vapours.

(c) Metallic ropes on inflammable-material vehicles

• During motion, tankers carrying petrol or other flammable liquids can accumulate static charge due to friction with the air and the road.
• A metallic rope or chain dragging on the ground provides a conducting path to earth, so charge is continuously discharged, reducing the risk of sparks and fire near inflammable materials.

(d) Bird on high-voltage line vs person on ground

• A bird sitting on a single bare high-voltage wire has both feet at almost the same potential, so there is practically no potential difference along its body; hence, almost no current flows through it.
• A person touching the same line while standing on the ground connects a very high potential (line) to nearly zero potential (earth), creating a large potential difference across the body; this drives a large current, which can be fatal.