Electromagnetism
Electromagnetism unifies electric and magnetic phenomena into a single theoretical framework, culminating in Maxwell's equations (1865). It explains everything from static charge interactions to light itself. The history runs from Coulomb's torsion balance (1785) through Faraday's field concept (1830s) to Maxwell's synthesis and Hertz's experimental confirmation of electromagnetic waves (1887). This skill covers electrostatics, magnetostatics, circuits, electromagnetic induction, Maxwell's equations, electromagnetic waves, and optics.
Agent affinity: maxwell (electromagnetism, Sonnet)
Concept IDs: phys-electric-charge-force, phys-ohms-law-circuits, phys-magnetic-fields, phys-electromagnetic-induction, phys-optics, phys-electromagnetic-spectrum
Electromagnetism at a Glance
| # | Topic | Key equations | Core idea | |---|---|---|---| | 1 | Coulomb's law & electric fields | F = kq_1 q_2/r^2, E = F/q | Charges create and respond to fields | | 2 | Gauss's law | Phi_E = Q_enc / epsilon_0 | Flux through closed surface = enclosed charge | | 3 | Electric potential | V = kq/r, E = -dV/dx | Potential energy per unit charge | | 4 | Capacitance | C = Q/V, C = epsilon_0 A/d | Stores charge and energy in fields | | 5 | DC circuits | V = IR, Kirchhoff's laws | Current, voltage, resistance relationships | | 6 | Magnetic fields & forces | F = qv x B, F = IL x B | Moving charges create and feel magnetic forces | | 7 | Biot-Savart & Ampere | dB = (mu_0/4pi)(I dL x r-hat)/r^2 | Current-carrying wires create B fields | | 8 | Faraday's law | EMF = -d(Phi_B)/dt | Changing flux induces voltage | | 9 | Maxwell's equations | Four equations unify all of E&M | Complete classical field theory | | 10 | EM waves | c = 1/sqrt(mu_0 epsilon_0) | Self-propagating E and B oscillations | | 11 | Optics | n_1 sin(theta_1) = n_2 sin(theta_2) | Light as electromagnetic wave |
Topic 1 — Coulomb's Law and Electric Fields
Coulomb's law: F = k |q_1 q_2| / r^2, where k = 1/(4 pi epsilon_0) = 8.99 * 10^9 N m^2/C^2. The force is along the line joining the charges: repulsive for like charges, attractive for unlike.
Electric field: E = F/q_test. The field at a point is the force per unit positive test charge. For a point charge Q: E = kQ/r^2, directed radially outward (positive Q) or inward (negative Q).
Superposition. The total electric field is the vector sum of fields from all individual charges. This principle underlies all electrostatics calculations.
Worked example. Two charges, q_1 = +3 muC at the origin and q_2 = -5 muC at x = 0.4 m. Find the electric field at x = 0.2 m.
Solution. At x = 0.2 m, the distance to q_1 is 0.2 m and to q_2 is 0.2 m.
E_1 = k|q_1|/r_1^2 = (8.99e9)(3e-6)/(0.04) = 6.74 * 10^5 N/C, pointing in the +x direction (away from positive charge).
E_2 = k|q_2|/r_2^2 = (8.99e9)(5e-6)/(0.04) = 1.124 * 10^6 N/C, pointing in the +x direction (toward negative charge).
E_total = E_1 + E_2 = 1.80 * 10^6 N/C in the +x direction.
Topic 2 — Gauss's Law
Statement: The electric flux through any closed surface equals the enclosed charge divided by epsilon_0:
Phi_E = integral of (E dot dA) over closed surface = Q_enc / epsilon_0
Power of Gauss's law. When the charge distribution has sufficient symmetry (spherical, cylindrical, or planar), Gauss's law reduces the field calculation to algebra.
Worked example. Find the electric field outside a uniformly charged sphere of total charge Q and radius R, at distance r > R from the center.
Solution. By spherical symmetry, E is radial and has the same magnitude at every point on a concentric Gaussian sphere of radius r. Gauss's law: E(4 pi r^2) = Q / epsilon_0. Therefore E = Q / (4 pi epsilon_0 r^2) = kQ/r^2.
This is identical to the field of a point charge Q at the center — the shell theorem for electrostatics.
When Gauss's law is not the right tool. When the charge distribution lacks high symmetry (e.g., a finite charged rod at a point off-axis), Gauss's law still holds but does not simplify. Use direct integration of Coulomb's law instead.
Topic 3 — Electric Potential and Voltage
Electric potential: V = U/q = kQ/r for a point charge. The potential is a scalar — no vector addition needed. The electric field is the negative gradient of potential: E = -grad(V).
Potential difference (voltage): Delta V = V_B - V_A = -integral from A to B of (E dot dL). This is what voltmeters measure.
Equipotential surfaces. Surfaces of constant V. The electric field is always perpendicular to equipotential surfaces. No work is done moving a charge along an equipotential.
Worked example. An electron (charge -e = -1.6 10^-19 C, mass 9.11 10^-31 kg) is accelerated from rest through a potential difference of 500 V. Find its final speed.
Solution. Energy conservation: q Delta V = (1/2)mv^2. Here q = e (magnitude of electron charge, since it gains energy moving from low to high potential).
v = sqrt(2e Delta V / m) = sqrt(2 1.6e-19 500 / 9.11e-31) = sqrt(1.758e14) = 1.326 * 10^7 m/s.
Check: v/c = 0.044, so non-relativistic treatment is justified (barely).
Topic 4 — Capacitance
Definition: C = Q/V. A capacitor stores charge Q when voltage V is applied. Unit: Farad (F) = C/V.
Parallel plate capacitor: C = epsilon_0 A / d, where A is plate area and d is plate separation.
Energy stored: U = (1/2)CV^2 = (1/2)Q^2/C = (1/2)QV.
Dielectrics. Inserting a dielectric material (dielectric constant kappa) between the plates increases capacitance: C = kappa epsilon_0 A / d. The dielectric reduces the effective electric field.
Series and parallel combinations:
- Parallel: C_total = C_1 + C_2 + ... (voltages equal, charges add)
- Series: 1/C_total = 1/C_1 + 1/C_2 + ... (charges equal, voltages add)
Note: This is the opposite pattern from resistors.
Topic 5 — DC Circuits
Ohm's law: V = IR. Resistance R = rho L / A, where rho is resistivity, L is length, A is cross-sectional area.
Power: P = IV = I^2 R = V^2/R.
Kirchhoff's laws:
- Junction rule (KCL): Sum of currents into a junction equals sum of currents out. (Conservation of charge.)
- Loop rule (KVL): Sum of voltage changes around any closed loop is zero. (Conservation of energy.)
Resistors in series: R_total = R_1 + R_2 + ... (currents equal, voltages add). Resistors in parallel: 1/R_total = 1/R_1 + 1/R_2 + ... (voltages equal, currents add).
Worked example. A circuit has a 12 V battery connected to three resistors: R_1 = 4 ohm and R_2 = 6 ohm in parallel, and that combination in series with R_3 = 5 ohm. Find the current from the battery.
Solution. Parallel combination: 1/R_12 = 1/4 + 1/6 = 3/12 + 2/12 = 5/12, so R_12 = 2.4 ohm. Total resistance: R_total = R_12 + R_3 = 2.4 + 5 = 7.4 ohm. Current: I = V/R_total = 12/7.4 = 1.62 A.
Worked example — Kirchhoff's laws. Two batteries (E_1 = 10 V with internal resistance r_1 = 1 ohm, and E_2 = 6 V with r_2 = 2 ohm) are connected in a loop with an external resistor R = 5 ohm. Find the current.
Solution. Apply KVL around the loop (assume current I flows clockwise from E_1): E_1 - I r_1 - IR - E_2 - I r_2 = 0. So 10 - I - 5I - 6 - 2I = 0. Thus 4 = 8I, giving I = 0.5 A.
Topic 6 — Magnetic Fields and Forces
Magnetic force on a moving charge: F = qv x B. The force is perpendicular to both v and B. It does no work (it changes direction, not speed).
Magnetic force on a current-carrying wire: F = IL x B, where L is the length vector along the wire in the direction of current.
Cyclotron motion. A charged particle in a uniform magnetic field moves in a circle: r = mv/(qB), period T = 2 pi m/(qB). The period is independent of speed (non-relativistic).
Worked example. A proton (m = 1.67 10^-27 kg, q = 1.6 10^-19 C) enters a 0.5 T magnetic field at 3 10^6 m/s perpendicular to the field. Find the radius of its circular path.*
Solution. r = mv/