AcademyGauss's Law
Academy
Charge on Conductors
Level 1 - Physics topic page in Gauss's Law.
Principle
In electrostatic equilibrium, conductor charge arranges itself so the electric field inside conducting material is zero.
Notation
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(\sigma\)
surface charge density
\(\mathrm{C\,m^{-2}}\)
\(q_{\mathrm{enc}}\)
charge enclosed by a Gaussian surface
\(\mathrm{C}\)
\(q,Q\)
cavity charge and conductor net charge
\(\mathrm{C}\)
\(\hat n\)
outward normal from the conductor
1
\(\epsilon_0\)
permittivity of free space
\(\mathrm{C^{2}\,N^{-1}\,m^{-2}}\)
Method
Derivation 1: Interior field of conducting material
Free charge in a conductor moves until there is no net electric force on charges inside the material.
Electric force
\[\vec F=q\vec E\]
Equilibrium condition
\[\vec F=0\quad\text{for free charge at rest}\]
Interior field
\[\vec E=0\quad\text{inside conducting material}\]
Derivation 2: Excess charge lies on surfaces
Place a Gaussian surface entirely inside the conducting material. Since \(\vec E=0\) there, its flux is zero.
Interior flux
\[\oint\vec E\cdot d\vec A=0\]
Gauss's law
\[q_{\mathrm{enc}}=\epsilon_0\oint\vec E\cdot d\vec A\]
No bulk excess
\[q_{\mathrm{enc}}=0\quad\text{within conducting material}\]
Derivation 3: Field just outside a conductor
Use a thin pillbox that crosses the surface. The inner cap is inside the conductor, where the field is zero.
Pillbox flux
\[\Phi_E=E_\perp A\]
Enclosed surface charge
\[q_{\mathrm{enc}}=\sigma A\]
Surface field
\[E_\perp=\frac{\sigma}{\epsilon_0}\]
Rules
These are conductor rules in electrostatic equilibrium.
Conductor interior
\[\vec E=0\quad\text{inside conducting material}\]
No bulk excess
\[q_{\mathrm{enc}}=0\quad\text{for a Gaussian surface in the material}\]
Surface field
\[\vec E_{\mathrm{outside}}=\frac{\sigma}{\epsilon_0}\hat n\]
Cavity charge
\[q_{\mathrm{inner}}=-q_{\mathrm{cavity}}\]
Outer surface
\[q_{\mathrm{outer}}=Q_{\mathrm{conductor}}+q_{\mathrm{cavity}}\]
Examples
Question
An isolated conducting sphere carries charge
\[+Q\]
Where is the excess charge?Answer
In electrostatic equilibrium, the excess charge lies on the outer surface. The field inside the conducting material is zero.
Checks
- \(\vec E=0\) applies inside conducting material, not automatically inside an empty cavity.
- Surface charge density can vary over a nonspherical conductor.
- The field just outside a conductor is normal to the surface.
- Cavity problems use net charge on each surface, not necessarily uniform charge density.