Difficulty: Intermediate | Prerequisites: Vector calculus basics (dot products, surface integrals), Coulomb's Law.
Gauss's Law is one of Maxwell's four equations and gives you a powerful shortcut for finding electric fields when the charge distribution has enough symmetry (spherical, cylindrical, or planar). Instead of integrating Coulomb's Law over every bit of charge, you choose a clever closed surface, relate the total flux through it to the enclosed charge, and read off the field. This topic also covers how conductors behave at electrostatic equilibrium, where charges live on the surface and the interior field is zero. You will use these ideas repeatedly when you reach capacitance, potential, and eventually electromagnetic waves.
Gauss's Law says the net electric flux through any closed surface equals the enclosed charge divided by ε₀. For symmetric charge distributions you can exploit this to find the electric field without integration. Inside a conductor at equilibrium the field is always zero, and all excess charge sits on the surface.
Electric flux (Φ_E)
The measure of how much electric field passes through a surface. Formally, Φ_E = ∮ E · dA. In simple terms, think of it as counting field lines piercing through a closed bag.
Gaussian surface
An imaginary closed surface you choose to apply Gauss's Law. It is not a physical object. Pick one whose shape matches the symmetry of your charge distribution so that E is constant over the surface or zero, making the integral trivial.
Gauss's Law
∮ E · dA = Q_enc / ε₀. The total electric flux out of any closed surface is proportional to the charge enclosed. In simple terms, the number of field lines leaving a region tells you how much charge is inside.
Linear charge density (λ)
Charge per unit length (C/m), used for problems with infinite or very long lines and cylinders of charge.
Volume charge density (ρ)
Charge per unit volume (C/m³), used for solid objects with charge spread throughout their bulk.
Electrostatic equilibrium
The state a conductor reaches when all charges have stopped moving. At equilibrium the electric field inside the conducting material is zero, and any net charge resides on the outer surface.
Induced charge
Charge that rearranges on a conductor's surface in response to a nearby charge or external field. The conductor's interior must remain field-free, so the surface charges redistribute to cancel any internal field.
The law works for any closed surface, but it is only useful for finding E when the geometry lets you factor the field out of the integral.
Three classic symmetries:
Spherical symmetry → use a concentric spherical Gaussian surface.
Cylindrical symmetry → use a coaxial cylindrical Gaussian surface.
Planar symmetry → use a pillbox (short cylinder straddling the plane).
Once you have the right surface, the integral reduces to E × (area of the surface) = Q_enc / ε₀, and you solve for E.
E = 0 everywhere inside the bulk of a conductor at equilibrium. If it were not zero, free charges would move until they cancelled the field.
All excess charge sits on the conductor's surface.
Just outside the surface, the field is perpendicular to the surface and has magnitude σ / ε₀ (where σ is the local surface charge density).
A solid metal cylinder of radius a with linear charge density λ_inner is surrounded by a cylindrical shell (inner radius b, outer radius c) with total linear charge density λ_shell.
Inside the solid cylinder (r < a): E = 0, because this is conducting material at equilibrium.
Between the conductors (a < r < b): Draw a cylindrical Gaussian surface of radius r and length L. Only λ_inner is enclosed. E(2πrL) = λ_inner · L / ε₀, giving E = λ_inner / (2πε₀r).
Inside the shell material (b < r < c): E = 0 (conductor).
Outside everything (r > c): Enclosed charge per length is λ_inner + λ_shell. If E = 0 for r > c, then λ_inner + λ_shell = 0, so λ_inner = −λ_shell.
When λ_inner > 0, the inner surface of the outer shell (at r = b) must carry induced charge −λ_inner to keep E = 0 inside the shell metal.
Solid non-conducting sphere of radius R, total charge Q, uniform volume charge density ρ = Q / (4πR³/3).
Choose a Gaussian sphere of radius r < R.
Enclosed charge: Q_enc = Q(r³ / R³), because the volume ratio of the inner sphere to the full sphere is r³ / R³.
Gauss's Law: E(4πr²) = Q(r³ / R³) / ε₀.
Solving: E = Qr / (4πε₀R³) = kQr / R³.
The field inside the sphere grows linearly with r, reaching its maximum at the surface.
Quantity | Expression |
|---|---|
Gauss's Law | ∮ E · dA = Q_enc / ε₀ |
Field outside infinite line/cylinder (r > R) | E = λ / (2πε₀r) |
Field inside conducting material | E = 0 |
Field inside uniformly charged sphere (r < R) | E = kQr / R³ |
Field outside uniformly charged sphere (r > R) | E = kQ / r² |
Coulomb constant | k = 1 / (4πε₀) ≈ 8.99 × 10⁹ N·m²/C² |
Permittivity of free space | ε₀ ≈ 8.85 × 10⁻¹² C²/(N·m²) |
The principle that a conductor shields its interior from external electric fields is the basis of the Faraday cage. This is why your car protects you in a lightning strike, and why sensitive electronics are housed in metal enclosures.
Students often think Gauss's Law only applies to symmetric situations. It always holds, but symmetry is needed to make it a practical tool for computing E.
A common error is trying to use the point-charge formula E = kQ/r² inside a uniformly charged non-conducting sphere. Inside the sphere, the enclosed charge is smaller than Q and the correct expression is E = kQr/R³.
Students sometimes forget that the field inside a conductor is zero because they confuse "conductor" with "insulator." A metal shell with charge on it still has E = 0 in the metal itself.
When finding induced charge on a conducting shell's inner surface, students sometimes think it depends on the shell's own net charge. It does not: the inner surface charge is fixed by the requirement that E = 0 inside the metal, regardless of λ_shell.
⚠️ The field inside a conductor at equilibrium is always zero. This appears repeatedly in multiple-choice and short-answer questions.
⚠️ For the uniformly charged sphere derivation, you must correctly identify the enclosed charge as Q(r³/R³), not Q. This is a classic five-mark short-answer question.
⚠️ For concentric conductors, the condition for zero external field is that the total enclosed charge is zero, giving the λ_inner = −λ_shell relationship.
⚠️ Induced charge on the inner surface of a surrounding shell is always equal and opposite to the enclosed charge, independent of the shell's own total charge.
True or false: The electric field inside a hollow conducting sphere is always zero, regardless of whether the sphere is charged.
Fill in the blank: For a Gaussian surface enclosing no net charge, the total electric flux through the surface is _______.
True or false: Inside a uniformly charged non-conducting sphere, the electric field increases linearly with distance from the centre.
Fill in the blank: If the electric field outside a pair of concentric cylindrical conductors is zero, the two linear charge densities must satisfy λ_inner + λ_shell = _______.
True or false: Gauss's Law can only be applied to symmetric charge distributions.
Answers: 1. True. 2. Zero. 3. True. 4. Zero. 5. False (it always holds; symmetry just makes it useful for calculating E).
Q: A solid metal cylinder has linear charge density λ_inner. A concentric cylindrical shell has total charge density λ_shell. If E = 0 for r > c (outside the shell), what is the relationship between λ_inner and λ_shell?
A: λ_inner = −λ_shell. By Gauss's Law, zero field outside requires zero total enclosed charge per unit length.
Q: What is the electric field inside a solid metal cylinder at electrostatic equilibrium?
A: E = 0. In any conductor at equilibrium, the internal field vanishes and excess charge resides on the surface.
Q: If λ_inner > 0, what sign is the induced charge on the inner surface (r = b) of the outer conducting shell?
A: Negative. A Gaussian surface inside the shell metal must enclose zero net charge, so the inner surface carries −λ_inner.
Q: Use Gauss's Law to find the electric field at distance r < R inside a uniformly charged non-conducting sphere of total charge Q and radius R.
A: E = Qr / (4πε₀R³) = kQr / R³. The enclosed charge is Q(r³/R³), and the Gaussian surface area is 4πr².
This connects directly to electric potential and capacitance: once you know E(r) from Gauss's Law, you integrate to find the potential difference, and from there you get the capacitance of spherical and cylindrical geometries (covered in the next set of notes). The conductor-at-equilibrium rules also reappear when you study shielding and grounding in circuits.
Gauss's Law, electric flux, Gaussian surface, enclosed charge, ε₀, permittivity of free space, Coulomb's Law, electric field inside conductor, electrostatic equilibrium, induced charge, surface charge density, linear charge density, volume charge density, concentric cylinders, concentric spheres, Faraday cage, PHYS 212, University Physics electricity and magnetism