Conductors

Assumptions

• Electrons cannot leave from the surface.
• Electrons can move freely within the material

When a conductor is in an electric field, the conductor creates an opposing field to nullify the effect of the field and reach static equilibrium.

$${\vec{E}}_{\text{net}}={\vec{E}}_{\text{ext}}+{\vec{E}}_{\text{int}}=0$$

To do this, charged particles move to a surface (electrons move against the field). This is called induction, and the charge forced to the surface is called induced charge.

Charged Conductors

If a conductor is forced to be charged, electrons are forced to surfaces, creating zero field everywhere in the material. They tend to clump up on sharp edges and needle-shaped surfaces.

Columb's Law

For a charged conductor with surface charge density of $\sigma$, the electric field close to the surface (outside) is:

$$\frac{\vec{E}\sigma }{{\epsilon }_0}{\vec{u}}_n$$

Electric field will be greater at the tips, since there is a higher charge density.

Hollow Conductors and Shielding

• Internal Field: In stable equilibrium, the electric field inside a hollow conductor’s cavity is zero (assuming no charges are inside).
• Internal Charges: If a charge $Q$ is placed inside, the inner surface accumulates $-Q$ (cancelling the field), and the outer surface gains $+Q$.
• Shielding: External fields cannot penetrate the conductor, creating a protected space (e.g., Faraday cages like cars).
• Potential: Because the internal field is zero, there is no potential difference anywhere in the conductor ($\Delta V=0$); it is an equipotential volume.