Source: UIUC Electricity, Magnetism, and Thermodynamics Final Examination
Tags: Coulomb's law, electric force, Gauss's law, electric flux, electric potential, conducting sphere, capacitors, series capacitors, parallel-plate capacitor, dielectrics, stored energy, charge distribution, surface charge density
Difficulty: Intermediate | Prerequisites: Calculus-based mechanics (PHYS 211 or equivalent), basic vector algebra, integral notation.
Electrostatics is the foundation of the entire electricity and magnetism sequence. It begins with how charges interact (Coulomb's law), builds to how electric fields behave in space (Gauss's law), and then connects fields to energy through electric potential. Capacitors are the first practical circuit element you meet, and understanding how they store charge and energy, especially when dielectrics are involved, is essential for everything that follows in AC circuits and electromagnetic waves. If you are comfortable with forces, fields, potentials, and capacitance, the rest of the course slots into place around those ideas.
Charged particles exert forces on each other proportional to the product of their charges and inversely proportional to the square of the distance between them. Gauss's law relates the total electric flux through a closed surface to the enclosed charge, regardless of the surface's shape or size. Capacitors store energy in the electric field between their plates, and inserting a dielectric or changing the circuit connection alters both the capacitance and the energy in predictable ways.
Coulomb's Law
The force between two point charges is F = kq₁q₂ / r², where k = 8.99 × 10⁹ N·m²/C². The force is attractive for opposite charges and repulsive for like charges. In simple terms, this is the electric equivalent of Newton's law of gravitation, but it can push as well as pull.
Electric Flux (Φ_E)
The total number of electric field lines passing through a surface, calculated as Φ_E = ∮ E · dA. Units are N·m²/C (or equivalently V·m). Think of it as a measure of "how much field" flows 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. In simple terms, the total flux depends only on the charge inside the surface, not on the surface's size or shape.
Electric Potential (V)
The electric potential energy per unit charge at a point in space: V = U/q. Measured in volts (J/C). Think of it as the "electrical height" at a point. Charges naturally move from high potential to low potential, just as objects roll downhill.
Capacitance (C)
The ratio of stored charge to voltage across a device: C = Q/V. Measured in farads (F). In simple terms, capacitance tells you how much charge a device can hold for each volt you apply.
Dielectric Constant (κ)
A dimensionless number indicating how much a dielectric material reduces the electric field between capacitor plates. κ is always greater than or equal to 1. Think of it as a multiplier for capacitance: inserting a dielectric with κ = 2 doubles the capacitance.
Surface Charge Density (σ)
Charge per unit area on a surface: σ = Q/A. Measured in C/m². In simple terms, this tells you how tightly packed the charges are on a given surface.
The electrostatic force between a proton and an electron in a hydrogen atom is calculated using F = ke² / r².
With e = 1.602 × 10⁻¹⁹ C and r = 5.29 × 10⁻¹¹ m (the Bohr radius), this gives F ≈ 8.2 × 10⁻⁸ N.
Although this force is tiny in absolute terms, it is roughly 10³⁹ times stronger than the gravitational force between the same two particles at the same distance.
Gauss's law is most useful when the charge distribution has spherical, cylindrical, or planar symmetry, because the symmetry lets you pull E out of the integral.
For an infinite line of charge with linear charge density λ, the natural Gaussian surface is a cylinder of radius r and length L centred on the line.
The enclosed charge is Q_enc = λL, so Φ_E = λL / ε₀.
Changing the radius of the cylinder does not change the enclosed charge. The flux remains the same regardless of the cylinder's radius.
For a point charge or a spherically symmetric distribution, use a spherical Gaussian surface.
In electrostatic equilibrium, the electric field inside a conductor is zero.
Because E = 0 everywhere inside, the potential is constant throughout the conductor's interior and equal to the potential on its surface.
For a solid conducting sphere of radius R carrying charge +Q, the potential at any point r < R is V = kQ/R (the same as at the surface), not kQ/r.
Capacitors in series all carry the same charge Q.
The equivalent capacitance is found from 1/C_eq = 1/C₁ + 1/C₂ + ...
For C₁ = 6 μF and C₂ = 12 μF in series: 1/C_eq = 1/6 + 1/12 = 3/12, so C_eq = 4 μF.
Connected to a 9 V battery: Q = C_eq × V = 4 μF × 9 V = 36 μC on each capacitor.
Capacitors in parallel share the same voltage but can carry different charges. C_eq = C₁ + C₂ + ...
When a charged capacitor is disconnected from the battery, the charge Q is fixed (it has nowhere to go).
Inserting a dielectric with constant κ increases the capacitance: C' = κC.
Since Q is fixed and U = Q²/(2C), the new energy is U' = Q²/(2κC) = U/κ.
With κ = 2, the stored energy decreases by a factor of 2.
The energy goes into pulling the dielectric slab into the gap (the field does work on the dielectric).
When the capacitor stays connected to the battery, the voltage V is fixed instead. Then U = CV²/2, and inserting the dielectric increases the energy by a factor of κ.
A point charge +q at the centre of a neutral conducting shell induces −q on the shell's inner surface and +q on its outer surface.
The surface charge density on the inner surface is σ = −q / (4πr_in²).
For q = 4 μC and r_in = 5 cm: σ = −4 × 10⁻⁶ / (4π × (0.05)²) ≈ −1.27 × 10⁻⁴ C/m².
Outside the shell, the system looks like a point charge +q at the origin.
Quantity | Formula |
|---|---|
Coulomb's law | F = kq₁q₂ / r² |
Coulomb constant | k = 8.99 × 10⁹ N·m²/C² |
Gauss's law | Φ_E = Q_enc / ε₀ |
Permittivity of free space | ε₀ = 8.85 × 10⁻¹² C²/(N·m²) |
Potential of point charge | V = kQ / r |
Potential inside conductor | V = kQ / R (constant, equal to surface value) |
Capacitors in series | 1/C_eq = 1/C₁ + 1/C₂ + ... |
Capacitors in parallel | C_eq = C₁ + C₂ + ... |
Energy in a capacitor | U = Q²/(2C) = CV²/2 = QV/2 |
Capacitance with dielectric | C' = κC |
Surface charge density | σ = Q / A |
Capacitors are everywhere: they smooth voltage in power supplies, store energy in camera flashes, and filter signals in audio equipment. The physics of dielectrics is the reason ceramic and electrolytic capacitors have different capacitance ratings for the same physical size. Gauss's law underpins the design of Faraday cages, which shield sensitive electronics (and people) from external electric fields.
Students often think that doubling the radius of a Gaussian surface around a line charge doubles the flux. It does not. Flux depends only on the enclosed charge, not on the surface dimensions.
Students frequently confuse the potential inside a conductor with zero. The field is zero inside, but the potential is constant and equal to the surface potential, which is generally not zero.
When a dielectric is inserted into a disconnected capacitor, students sometimes assume the energy increases because the capacitance increases. The energy decreases because Q is fixed and U = Q²/(2C): a larger C in the denominator gives a smaller U.
In series circuits, students sometimes try to add capacitances directly. Series capacitances add reciprocally; parallel capacitances add directly.
⚠️ Coulomb's law calculations require careful attention to units and powers of ten. Off-by-one exponent errors are the most common way to lose marks.
⚠️ Gauss's law questions almost always test whether you know that flux depends on enclosed charge, not on the Gaussian surface geometry. Read the question for what is changing (the surface) versus what is staying the same (the charge).
⚠️ Know the difference between a capacitor disconnected from the battery (Q fixed) and one still connected (V fixed). This distinction determines whether energy goes up or down when you insert a dielectric.
⚠️ Electric field lines point from high potential to low potential. The exam true/false section tests this: the statement that field lines point from low to high potential is false.
True or false: The electric flux through a closed surface depends on the size and shape of that surface.
Fill in the blank: Inside a solid conducting sphere in electrostatic equilibrium, the electric field is ______.
True or false: Capacitors in series all carry the same charge.
Fill in the blank: Inserting a dielectric into a disconnected capacitor ______ (increases/decreases) the stored energy.
True or false: The capacitance of a parallel-plate capacitor increases when the distance between the plates is increased.
Answers: 1. False (depends only on enclosed charge). 2. Zero. 3. True. 4. Decreases. 5. False (C = ε₀A/d, so larger d means smaller C).
Q: A hydrogen atom has a proton and electron separated by r = 5.29 × 10⁻¹¹ m. What is the magnitude of the electric force between them?
A: F = ke²/r² = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)² / (5.29 × 10⁻¹¹)² ≈ 8.2 × 10⁻⁸ N.
Q: An infinite line charge has linear density λ. If you double the radius of the cylindrical Gaussian surface, what happens to the flux?
A: The flux remains the same. Gauss's law says Φ = Q_enc/ε₀, and doubling the radius does not change the enclosed charge.
Q: What is the electric potential inside a conducting sphere of radius R carrying charge +Q?
A: V = kQ/R at every point inside the sphere, the same as at the surface.
Q: Two capacitors (6 μF and 12 μF) are in series across 9 V. What charge is stored on each?
A: C_eq = 4 μF. Q = C_eq × V = 36 μC on each capacitor.
Q: A charged parallel-plate capacitor is disconnected from the battery and a dielectric (κ = 2) is inserted. What happens to the stored energy?
A: The energy is halved. With Q fixed, U' = U/κ = U/2.
Q: A point charge +q = 4 μC sits at the centre of a neutral conducting shell (inner radius 5 cm). What is the surface charge density on the inner surface?
A: The inner surface carries −q = −4 μC. σ = −4 × 10⁻⁶ / (4π(0.05)²) ≈ −1.27 × 10⁻⁴ C/m².
Gauss's law reappears in magnetism as one of Maxwell's equations (the magnetic version states that the net magnetic flux through any closed surface is zero, reflecting the absence of magnetic monopoles). Capacitor energy storage connects directly to LC and RLC oscillations, where energy shuttles between the capacitor's electric field and the inductor's magnetic field. The concept of electric potential is the bridge to understanding voltage in circuits, EMF in batteries, and potential difference across resistors.
Coulomb force, electrostatic force, point charge, inverse square law, electric flux, Gaussian surface, cylindrical symmetry, spherical symmetry, Gauss's law applications, electric potential energy, voltage, equipotential surface, conductor in electrostatic equilibrium, capacitance formula, series and parallel capacitors, equivalent capacitance, dielectric material, dielectric constant, permittivity, energy stored in capacitor, capacitor energy with dielectric, concentric spherical shell, induced charge, surface charge density, Faraday cage