Course: University Physics 212: Electricity and Magnetism | University of Illinois at Urbana-Champaign
Source: Final Assessment review material
Difficulty: Intermediate Prerequisites: PHYS 211 (mechanics), comfort with vectors, basic calculus (integration over simple volumes and surfaces)
Tags: Coulomb's law, electric field, point charge, superposition, Gauss's law, Gaussian surface, electric flux, charge distribution, conducting sphere, conducting shell, infinite slab, infinite line charge, field lines, electrostatics, PHYS 212
Electrostatics is the study of stationary charges and the forces and fields they produce. It forms the first major pillar of PHYS 212 and underpins nearly everything that follows: electric potential, capacitors, and circuits all rest on these ideas. You should already be comfortable with vectors, Newton's laws, and basic integration. If Coulomb's law or the concept of a field feels unfamiliar, start here before moving on to potential or circuits.
Charges exert forces on each other that fall off as 1/r². The electric field is the force per unit charge and depends on the geometry of the charge distribution. Gauss's law relates the total electric flux through a closed surface to the enclosed charge, and it becomes a powerful shortcut when the charge distribution has enough symmetry.
Coulomb's law
The force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them: F = kq₁q₂/r². In simple terms, double the distance and the force drops to one quarter.
Electric field (E)
The electric field at a point in space is the force a positive test charge would experience per unit charge: E = F/q. Think of it as the "influence map" a charge creates around itself.
Electric field lines
Visual representations of the electric field. Lines point away from positive charges and toward negative charges. Where lines are closer together, the field is stronger.
Electric flux (Φ_E)
The "flow" of electric field through a surface: Φ_E = ∮ E · dA. In simple terms, it measures how much field passes through a given area.
Gauss's law
The net electric flux through any closed surface equals the enclosed charge divided by ε₀: Φ_E = Q_enc / ε₀. This is always true, but it is only useful as a calculation tool when the geometry has high symmetry.
Gaussian surface
An imaginary closed surface chosen to exploit symmetry so that E · dA is constant or zero over each part of the surface. The shape must match the symmetry of the charge distribution.
Linear charge density (λ)
Charge per unit length (C/m). Used for wires, lines of charge, and other one-dimensional distributions.
Volume charge density (ρ)
Charge per unit volume (C/m³). Used for solid objects with charge distributed throughout their volume.
Electrostatic equilibrium
A conductor is in electrostatic equilibrium when there is no net motion of charge. In this state, the electric field inside the conductor is zero and any excess charge sits on the surface.
The electrostatic force between two point charges q₁ and q₂ separated by distance d is:
F = kq₁q₂ / d²
k = 8.99 × 10⁹ N·m²/C² (also written as 1/(4πε₀))
Because force scales as 1/d², doubling the separation reduces the force to one quarter of its original value. Halving the separation quadruples the force.
The force is attractive between unlike charges and repulsive between like charges.
Point charge: E = kQ/r². Field points radially outward from a positive charge and radially inward toward a negative charge.
Infinite line of charge (linear density λ): E = λ/(2πε₀r). The field falls off as 1/r, not 1/r². This comes directly from applying Gauss's law with a cylindrical Gaussian surface.
Infinite slab of thickness d, uniform charge density ρ: Inside the slab, E grows linearly with distance z from the centre plane: E = ρz/ε₀. The Gaussian surface here is a flat "pillbox" centred on the midplane.
Lines originate on positive charges and terminate on negative charges.
Near a positive point charge, lines point radially outward.
Lines never cross. The density of lines indicates field strength.
For uniform fields (e.g. between large parallel plates), lines are parallel and evenly spaced.
Gauss's law is: ∮ E · dA = Q_enc / ε₀
It is always valid, but only useful for easy calculation when E is constant over parts of the Gaussian surface and perpendicular (or parallel) to it.
Symmetry types and their matching Gaussian surfaces:
Spherical symmetry (point charge, uniformly charged sphere) → spherical Gaussian surface
Cylindrical symmetry (infinite line, infinite cylinder) → cylindrical Gaussian surface
Planar symmetry (infinite plane, infinite slab) → pillbox (flat cylinder)
A charged cube does not have sufficient symmetry for Gauss's law to yield a simple expression for E. You can still compute total flux through a surface enclosing the cube, but you cannot pull E out of the integral.
E = 0 everywhere inside the conducting material.
All excess charge resides on the surface.
When a charge +Q sits at the centre of a neutral conducting shell (inner radius b, outer radius c):
The inner surface of the shell acquires charge −Q (to make E = 0 inside the conductor).
The outer surface acquires charge +Q (to keep the shell neutral overall).
In the gap between the central charge and the shell (a < r < b), the field is the same as for a point charge: E = kQ/r².
Inside the shell material (b < r < c), E = 0.
Outside the shell (r > c), E = kQ/r², exactly as if all the charge were at the centre.
Quantity | Formula | Notes |
|---|---|---|
Coulomb's law | F = kq₁q₂/r² | k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C² |
Point charge field | E = kQ/r² | Radial, 1/r² dependence |
Infinite line field | E = λ/(2πε₀r) | 1/r dependence |
Infinite slab (inside) | E = ρz/ε₀ | Linear in z from centre plane |
Gauss's law | Φ_E = Q_enc/ε₀ | ε₀ = 8.85 × 10⁻¹² C²/(N·m²) |
Gauss's law is the reason coaxial cables work: the outer conductor shields the inner conductor's field, so signals stay contained and external interference is blocked. Electrostatic shielding in general (Faraday cages, shielded enclosures) relies on the same conductor-in-equilibrium physics.
Students often assume the electric field from an infinite line of charge falls off as 1/r². It does not. The field falls as 1/r because the line extends infinitely, contributing field from all along its length.
Students sometimes think Gauss's law can always be used to calculate E directly. It can always compute total flux, but it only gives E simply when the geometry has enough symmetry to pull E out of the integral.
A common error is choosing a Gaussian surface that does not match the charge distribution's symmetry, for instance, using a sphere around a cube of charge to find E. The flux calculation still holds, but E varies over the surface, so you cannot solve for it.
Students forget that for a neutral conducting shell surrounding a point charge, the field outside the shell is nonzero. The shell redistributes charge on its surfaces; it does not eliminate the field outside.
⚠️ Coulomb's law inverse-square problems are classic exam questions. Know that doubling distance quarters the force.
⚠️ For Gauss's law problems, the first decision is always: does this geometry have symmetry? If not, Gauss's law will not give you E directly.
⚠️ Conducting shell problems (charge induction on inner/outer surfaces) appear regularly in PHYS 212 finals. Memorise the logic: inner surface charge cancels the enclosed charge; outer surface carries whatever is left.
⚠️ Know the field dependence for each geometry: 1/r² for a point charge, 1/r for an infinite line, constant for an infinite plane, linear in z for inside a slab.
True or false: If you double the distance between two point charges, the force between them halves.
False. The force drops to one quarter (inverse-square law).
Fill in the blank: The electric field from an infinite line of charge is proportional to ______.
1/r
True or false: Gauss's law can always be used to directly calculate the electric field magnitude.
False. It always gives total flux, but it only gives E directly when there is sufficient symmetry.
True or false: Inside a conductor in electrostatic equilibrium, the electric field is zero.
True.
Fill in the blank: The charge induced on the inner surface of a neutral conducting shell surrounding a point charge +Q is ______.
−Q
Q: Two point charges are separated by distance d. If the distance is doubled to 2d, what happens to the electrostatic force?
A: The force decreases to one quarter of its original value. F ∝ 1/d², so doubling d gives F/(2²) = F/4.
Q: An infinite line of charge has linear charge density λ. How does the electric field magnitude E depend on the perpendicular distance r from the line?
A: E ∝ 1/r. From Gauss's law with a cylindrical surface: E = λ/(2πε₀r).
Q: You have a charged cube of side a. Can you use Gauss's law to easily find E at a distance R > a from the cube?
A: No. The cube lacks sufficient symmetry. You could compute the total flux through a closed surface, but E is not constant over any simple Gaussian surface around a cube.
Q: A point charge +Q is at the centre of a neutral conducting spherical shell. What is the charge on the inner surface of the shell?
A: −Q. This charge is induced so that the field inside the conducting material is zero.
Q: How does the electric field inside an infinite uniformly charged slab vary with distance z from the centre plane?
A: E is proportional to z (E = ρz/ε₀). It grows linearly from zero at the centre to a maximum at the surface.
Q: Which of the following describes field lines near a positive point charge: parallel lines, radially inward, closed loops, or radially outward?
A: Radially outward. Field lines originate on positive charges.
This material connects directly to electric potential (Part 2 of these notes), because potential is defined as the work done against the electric field per unit charge. Understanding how E behaves for different geometries is essential for computing potential differences.
Gauss's law reappears in magnetism as one of Maxwell's equations: the magnetic flux through any closed surface is always zero (no magnetic monopoles). The mathematical structure is identical.
The conductor-in-equilibrium results also underpin capacitor physics. A parallel-plate capacitor is, at heart, two conductors with opposite surface charges and a uniform field between them.
Coulomb's law, electric force, inverse square law, electric field, field lines, superposition principle, Gauss's law, Gaussian surface, electric flux, charge density, linear charge density, volume charge density, surface charge density, conducting sphere, conducting shell, Faraday cage, electrostatic shielding, electrostatic equilibrium, pillbox surface, cylindrical Gaussian surface, spherical symmetry, PHYS 212 UIUC, E&M final review