Source: University Physics: Elec & Mag, UIUC
Tags: Gauss's law, electric flux, Gaussian surface, conductors, electric potential energy, work-energy theorem, conservative force, PHY 212
Difficulty: Intermediate Prerequisites: Coulomb's law and electric fields study notes, dot product, surface integrals (conceptual), work and energy from mechanics.
Gauss's law is a powerful restatement of Coulomb's law that lets you bypass difficult integrals whenever the charge distribution has enough symmetry (spherical, cylindrical, or planar). Instead of integrating over every tiny charge element, you draw an imaginary closed surface, count how much charge sits inside, and read off the electric field. This topic also introduces how conductors behave in electrostatics, which matters for every capacitor problem later. Electric potential energy then connects the electrostatic force to the energy framework you already know from mechanics: if the force is conservative, there is a potential energy function, and changes in that function equal the negative of the work done by the field.
Gauss's law says the total electric flux through any closed surface equals the enclosed charge divided by ε₀. When symmetry lets you pull E out of the integral, you get E directly. Electric potential energy is the energy stored in a configuration of charges; it changes by ΔU = −W_conservative, and for two point charges U(r) = k·q₁·q₂ / r.
Electric Flux (Φ)
The integral of the electric field dotted with the outward area element over a surface: Φₛ = ∫ E · dA. Think of it as counting how many field lines pass through a surface.
Gauss's Law
∮ E · dA = Q_enclosed / ε₀. The net electric flux through any closed surface depends only on the total charge enclosed, not on the shape of the surface or any charges outside it.
Gaussian Surface
An imaginary closed surface chosen to exploit symmetry so that E is constant over the surface (or zero, or perpendicular to dA on certain parts). It is a mathematical tool, not a physical object.
Conductor (electrostatic)
A material in which charges are free to move. At equilibrium: E = 0 everywhere inside, excess charge resides entirely on the surface, and the surface is an equipotential.
Electric Potential Energy (U)
The energy associated with the configuration of charges. For a conservative force, ΔU = −W_conservative. Think of it as the electric equivalent of gravitational potential energy: it tells you how much work the field can do (or has had done against it) as charges move.
Conservative Force
A force whose work depends only on the start and end positions, not the path. The electrostatic force is conservative, which is why a potential energy function exists for it.
Φₛ = ∫ E · dA, where dA is the outward-pointing area vector of a small surface element.
Three cases for the dot product:
E parallel to dA (outward): E · dA > 0 (positive flux).
E has a component along dA: E · dA = E · dA · cos θ.
E perpendicular to dA: E · dA = 0 (no flux through that patch).
Positive flux means net field lines leaving the surface; negative means net lines entering.
∮ (closed surface) E · dA = Q_enclosed / ε₀.
The shape of the closed surface does not matter. As long as Q_enclosed is the same, the total flux is the same.
Charges outside the surface contribute zero net flux (their field lines enter and exit in equal measure).
Only practical when symmetry lets you pull E outside the integral, giving: E · A = Q_enclosed / ε₀, so E = Q_enclosed / (A · ε₀).
Choose the Gaussian surface so that:
E is constant over the surface and equal to the value you want.
E · dA = 0 on any remaining parts (field perpendicular to the surface normal there).
Spherical (point charge, uniformly charged sphere): Gaussian surface is a concentric sphere, A = 4πr². Result: E = Q_enc / (4πε₀ · r²).
Cylindrical (infinite line charge, long charged cylinder): Gaussian surface is a coaxial cylinder, A_curved = 2πr · L. Result: E = λ / (2πε₀ · r).
Planar (infinite charged plane): Gaussian surface is a "pillbox" straddling the plane, A = 2πr² (both flat faces). Result: E = σ / (2ε₀).
A cube does not obey any of these symmetries, so Gauss's law is not a shortcut there.
E = 0 everywhere inside a conductor (charges rearrange until they cancel any internal field).
Excess charge sits only on the surface.
The surface of a conductor is an equipotential.
E at the surface is perpendicular to the surface.
To prove E = 0 inside, draw a Gaussian surface entirely within the metal: Q_enclosed = 0, so E = 0.
A point charge +Q at the centre of a neutral conducting shell with inner radius r₁ and outer radius r₂.
Region r < r₁: E = kQ / r² (only +Q enclosed).
Region r₁ < r < r₂ (inside the metal): E = 0.
Region r > r₂: The shell's outer surface carries +Q (induced), so E = kQ / r² again.
Induced charge of −Q appears on the inner surface to ensure E = 0 inside the conductor.
ΔU = U_final − U_initial = −W_conservative.
Nature drives systems toward lower potential energy.
The electrostatic force is conservative, so a potential energy function exists.
U(r) = k · q₁ · q₂ / r.
Reference: U(∞) = 0 (zero potential energy when the charges are infinitely far apart).
Same-sign charges: U > 0 (you had to do work to bring them together).
Opposite-sign charges: U < 0 (the field did the work).
For three charges (Q₁, Q₂, q) with Q₁ and Q₂ initially separated by d, and q brought from infinity to distance d from both:
ΔU = (1/4πε₀) · [q·Q₁/d + q·Q₂/d].
The total potential energy of the configuration is the sum of the energies of every distinct pair.
Given charges +Q at some position and −2Q at another along the x-axis, find where a third charge +q can be placed so that the total change in potential energy is zero.
Set up: ΔU = −W_Qq − W_{−2Q,q} = 0, which gives a condition relating the distances d₁ and d₂ to the two existing charges.
Solution: d₂ = 2d₁, yielding two possible locations along the axis.
Quantity | Formula |
|---|---|
Electric flux | Φ = ∫ E · dA |
Gauss's law | ∮ E · dA = Q_enc / ε₀ |
E, spherical symmetry | E = Q_enc / (4πε₀ r²) |
E, cylindrical symmetry | E = λ / (2πε₀ r) |
E, planar symmetry | E = σ / (2ε₀) |
Potential energy (two point charges) | U = k q₁ q₂ / r |
Change in potential energy | ΔU = −W_conservative |
Gauss's law explains why a Faraday cage works: inside a closed conductor, the electric field is zero regardless of what is happening outside. This principle protects sensitive electronics from external fields and is the reason your car is a reasonably safe place during a lightning storm. Electric potential energy is the basis for understanding how charged-particle accelerators work, and it governs the energy stored in every capacitor and battery.
Students often think Gauss's law only works for certain shapes of surface. The law is always true for any closed surface. The restriction is practical: you can only solve for E easily when the symmetry lets you pull it out of the integral.
Confusing "Q enclosed" with "total charge in the problem." Only the charge inside the Gaussian surface matters for flux; charges outside contribute zero net flux.
Assuming that if E = 0 at a point, there must be no charge nearby. E = 0 inside a conductor at equilibrium, but charge is sitting right there on its surface.
Mixing up the sign of work and the sign of ΔU. If the electric field does positive work on a charge, the potential energy decreases (ΔU < 0), not increases.
⚠️ Choosing the right Gaussian surface is the core exam skill. Know the three standard geometries and their area formulas cold.
⚠️ Conductor problems (induced charge, E inside/outside a shell) appear frequently. Walk through the logic: E = 0 inside metal, therefore Q_enclosed by a surface inside the metal = 0, therefore induced charge on the inner surface = −Q.
⚠️ ΔU = −W_conservative is the sign convention that trips people up. Practice stating in words: "The field does positive work, so PE decreases."
⚠️ For potential energy of a system of charges, count every unique pair exactly once.
Q: True or false – The electric flux through a closed surface depends on the shape of that surface.
A: False. It depends only on the total charge enclosed.
Q: Fill in the blank – Inside a conductor at electrostatic equilibrium, the electric field is ___.
A: Zero.
Q: True or false – If two positive charges are brought closer together, their electric potential energy decreases.
A: False. It increases (you must do work against the repulsive force).
Q: What is the reference point for electric potential energy of two point charges?
A: U = 0 when the charges are infinitely far apart.
Q: A spherical Gaussian surface of radius 0.5 m encloses a total charge of +4 μC. What is the electric flux through the surface?
A: Φ = Q_enc / ε₀ = 4 × 10⁻⁶ / 8.85 × 10⁻¹² ≈ 4.52 × 10⁵ N·m²/C.
Q: An infinite plane carries a uniform surface charge density σ = 6 nC/m². What is the magnitude of the electric field near the surface?
A: E = σ / (2ε₀) = 6 × 10⁻⁹ / (2 × 8.85 × 10⁻¹²) ≈ 339 N/C.
Q: A neutral conducting shell surrounds a point charge +Q. What charge appears on the shell's inner surface? Outer surface?
A: Inner surface: −Q. Outer surface: +Q. The shell is neutral overall, so the induced charges must sum to zero.
Q: Two protons are released from rest, separated by 1 × 10⁻¹⁰ m. Describe what happens to their potential energy and kinetic energy as they move apart.
A: Potential energy decreases (positive charges repelling, field does positive work). Kinetic energy increases by the same amount (energy conservation). Total energy is constant.
Gauss's law and conductors connect directly to capacitance: every parallel-plate, cylindrical, and spherical capacitor calculation starts by using Gauss's law to find E between the plates. Electric potential energy connects to electric potential (V = U/q), which is the next topic; V is the per-unit-charge version of U, and it is the quantity you measure with a voltmeter.
Gauss's law, electric flux, Gaussian surface, closed surface, spherical symmetry, cylindrical symmetry, planar symmetry, conductor, induced charge, Faraday cage, electrostatic equilibrium, electric potential energy, conservative force, work-energy theorem, point charge potential energy, PHY 212 midterm 1, UIUC physics