Review / Chapter 3 — Conductors

Properties of Conductors

Chapter 3 — Conductors

A conductor is a material containing a large population of free charges (electrons in a metal, ions in salty water). In electrostatic equilibrium these free charges have rearranged themselves so that nothing is accelerating. That single requirement — no free charge feels a net force — locks in a whole family of powerful consequences.

The five properties

In electrostatic equilibrium:

  1. The field vanishes inside: \(\vec{E} = 0\) everywhere in the bulk. If it weren't, free charges would feel a force and keep moving — not equilibrium.
  2. Conductors are equipotentials: \(V = \text{const}\) throughout. For any two points connected by a path inside the conductor, \(\Delta V = -\int \vec{E}\cdot d\vec{s} = 0\).
  3. External field is perpendicular to the surface: any tangential component would push surface charges along the surface.
  4. No bulk charge: \(\rho = \epsilon_0\,\vec\nabla\cdot\vec{E} = 0\) inside (since \(\vec{E}=0\)).
  5. All net charge lives on the surface: consequence of (4) combined with charge conservation.

Induced charge and screening

Drop a neutral conductor into an external field. The free charges drift until they cancel that field inside the conductor. The result is a non-uniform surface charge — negative on the side facing positive external charges, positive on the opposite side. From outside, the conductor looks like it has acquired a dipole (and higher multipoles).

Neutral conducting sphere placed in an external field, showing induced surface charge
A neutral conducting sphere in a uniform external field. Induced surface charges (negative on the left, positive on the right) cancel the field inside. The external field lines meet the surface perpendicularly.

Grounded vs. isolated

An isolated conductor has a fixed total charge \(Q\); its potential adjusts. A grounded conductor is connected to a huge reservoir (the Earth) held at \(V = 0\); charge flows on or off freely to maintain that potential. Most image-charge problems use grounded conductors.

Cavities inside conductors

If the conductor has a hollow cavity and no charge is placed in the cavity, the field in the cavity is exactly zero — regardless of what happens outside. If you put a charge \(q\) inside the cavity, an induced charge \(-q\) appears on the cavity wall and \(+q\) redistributes on the outer surface (see the Faraday cage page for more).

Practice Problems

Problem 1easy
A solid metal sphere carries net charge \(Q\). Where is the charge located, and what is the field at the center?
Hint
Properties 1 and 5.
Solution

All \(Q\) sits on the outer surface (no bulk charge). The field everywhere inside — including the center — is zero.

Answer: All charge on the surface; \(\vec{E}(0) = 0\).

Problem 2medium
Two thin concentric conducting shells have radii \(R_1 < R_2\) and carry net charges \(Q_1\) and \(Q_2\). Find the charge on each of the four surfaces (inner and outer of each shell) and the field in all three regions.
Hint
Enforce \(\vec{E}=0\) inside each conductor using a Gaussian sphere.
Solution

Inside shell 1 (\(r < R_1\)): by symmetry and \(\vec{E}=0\) inside the conductor at \(r=R_1^-\), the enclosed charge must be zero, so the inner surface of shell 1 carries 0 and the outer surface carries \(Q_1\). Between shells (\(R_1 < r < R_2\)): \(\vec{E} = \frac{kQ_1}{r^2}\hat r\). Inside shell 2 the field is again zero, so a Gaussian sphere at \(r = R_2^-\) requires enclosed charge \(0\), forcing \(-Q_1\) on the inner surface of shell 2. Charge conservation on shell 2 puts \(Q_1 + Q_2\) on its outer surface. Outside (\(r > R_2\)): \(\vec{E} = \frac{k(Q_1+Q_2)}{r^2}\hat r\). Inside shell 1: \(\vec{E}=0\).

Answer: surfaces \((0,\ Q_1,\ -Q_1,\ Q_1+Q_2)\) from innermost to outermost.

Problem 3medium
A neutral conductor has a cavity containing a point charge \(+q\). An additional charge \(+q_0\) sits far outside. What is the force on the inside charge? On the outside charge, approximately?
Hint
The conductor screens the cavity from the outside world.
Solution

The cavity wall develops induced charge \(-q\); the outside gets \(+q\). If the cavity is spherical and the charge is at its center, the induced \(-q\) is uniform and exerts zero net force on \(q\). Even off-center, the only field inside the cavity comes from charges in the cavity — so the force on \(q\) from "outside" is zero: complete screening. The outside charge \(q_0\) sees a distant conductor with net surface charge \(+q\) on the outside (since \(d \gg R\), it looks like a point charge \(+q\)), giving a Coulomb force \(\approx \frac{1}{4\pi\epsilon_0}\frac{q\,q_0}{d^2}\).

Answer: force on inside charge is 0 (spherical cavity, charge at center); outside charge feels \(\approx kqq_0/d^2\).

Problem 4easy
True or false: if a conductor has zero net charge, then the surface charge density is zero everywhere on it.
Hint
Neutral overall doesn't mean neutral locally.
Solution

False. A neutral conductor in an external field has equal amounts of induced \(+\) and \(-\) surface charge on opposite sides. Locally \(\sigma \ne 0\); globally \(\oint \sigma\,dA = 0\).

Answer: False.