Introduction
This chapter builds the classical theory of electric and magnetic fields, culminating in Maxwell’s equations — four equations that unify electricity, magnetism, and light into a single framework. Its logical order moves from static charges (electrostatics) to static currents (magnetostatics) to the coupling between changing fields (electromagnetic induction), and finally to the complete, time-dependent theory.
1. Electric field and Gauss’s law
An electric field, E, at a point is the force per unit positive test charge that would be experienced there: E = F/q. Field lines point away from positive charges and into negative charges, and their density represents field strength.
Gauss’s law, ∮E·dA = Q_enclosed/ε₀, states that the total electric flux through any closed surface is proportional to the charge enclosed by that surface. Its power is computational: for situations with high symmetry (a sphere, an infinite line, or an infinite plane of charge), Gauss’s law lets you find the field with simple algebra instead of a difficult direct integration over every charge element.
2. Electric potential, Poisson’s and Laplace’s equations
Electric potential, V, is potential energy per unit charge, related to the field by E = −∇V — the field points in the direction of steepest potential decrease. Potential is often easier to work with than the field itself because it is a scalar.
Poisson’s equation, ∇²V = −ρ/ε₀, relates the potential at every point to the local charge density ρ. In charge-free regions, this reduces to Laplace’s equation, ∇²V = 0. These equations, combined with boundary conditions (values of V on surrounding conductors), let you find the potential (and hence the field) in complex geometries such as capacitors and conductors of arbitrary shape.
3. Dielectrics, polarization, and capacitance
A dielectric is an insulating material that, when placed in an electric field, develops an internal separation of positive and negative charge (polarization) without free charges flowing — the induced dipoles partially oppose the external field, reducing the net field inside the material. A capacitor stores charge and energy; its capacitance, C = Q/V, increases when a dielectric is inserted between its plates, because the dielectric reduces the potential difference needed to hold a given charge.
4. Magnetic field of moving charges and Ampère’s law
A moving charge, unlike a stationary one, produces a magnetic field, described by the Biot–Savart law for a current element. Ampère’s law, ∮B·dl = μ₀I_enclosed, plays the same computational role for magnetic fields with high symmetry (an infinite wire, a solenoid) that Gauss’s law plays for electric fields — it relates the circulation of B around a closed loop to the current passing through that loop.
5. Magnetic properties of matter
Materials respond to an applied magnetic field in three broad categories, depending on their atomic magnetic moments: diamagnetic materials are weakly repelled (induced moments oppose the field); paramagnetic materials are weakly attracted (existing atomic moments partially align with the field but are randomised by thermal motion); ferromagnetic materials (iron, cobalt, nickel) show strong, persistent magnetisation because neighbouring atomic moments align cooperatively within regions called domains, which is why they can form permanent magnets.
6. Faraday’s law of induction and AC circuits
Faraday’s law, EMF = −dΦ_B/dt, states that a changing magnetic flux through a circuit induces an electromotive force (voltage) — the basis of electric generators, transformers, and inductors. The negative sign is Lenz’s law: the induced current always opposes the change that caused it, consistent with energy conservation.
In RLC circuits (resistor, inductor, capacitor) driven by an alternating current, the resistor’s voltage stays in phase with current, the inductor’s voltage leads current by 90°, and the capacitor’s voltage lags by 90°. At the circuit’s resonant frequency, the inductive and capacitive reactances cancel, current is maximised, and the circuit behaves purely resistively — the principle used to tune a radio receiver to a specific station.
7. Maxwell’s equations, the Poynting vector, and potentials
Maxwell’s four equations summarise all of classical electromagnetism: Gauss’s law for E, Gauss’s law for B (stating magnetic monopoles do not exist — field lines always close on themselves), Faraday’s law, and the Ampère–Maxwell law (Ampère’s law extended with Maxwell’s crucial addition of a “displacement current” term, needed for consistency and which predicts that a changing electric field itself generates a magnetic field). Together, these equations show that a self-sustaining electromagnetic wave can propagate through empty space at speed c — unifying electricity, magnetism, and optics.
The Poynting vector, S = (1/μ₀)E×B, gives the direction and intensity of energy flow carried by an electromagnetic wave — essential for understanding how energy is radiated by antennas or carried by sunlight. Scalar and vector potentials (V and A) are mathematical constructions from which E and B can be derived (E = −∇V − ∂A/∂t, B = ∇×A); they simplify many calculations and become essential in the quantum treatment of electromagnetism.
Key Takeaways
- Gauss’s law and Ampère’s law exploit symmetry to compute fields easily; potential (a scalar) is often more convenient than field (a vector) and obeys Poisson’s/Laplace’s equations.
- Dielectrics increase capacitance by polarizing and partially cancelling the applied field; magnetic materials are classified by how their atomic moments respond (diamagnetic, paramagnetic, ferromagnetic).
- Faraday’s law (changing flux induces EMF) and Lenz’s law (the induced effect opposes the change) underlie generators, transformers, and AC circuit behaviour, including resonance.
- Maxwell’s equations unify electricity and magnetism and predict electromagnetic waves travelling at the speed of light; the Poynting vector describes the energy such waves carry.