A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The Lorentz force law F = q(E + v × B) unifies electric and magnetic phenomena. Combined with Maxwell's equations (Gauss's law, no magnetic monopoles, Faraday's law, Ampere-Maxwell law), it forms the complete framework of classical electromagnetism.
Revisit Gauss's law, Faraday's law, and Ampere's law as four equations governing E and B. Solve a problem involving both electric and magnetic forces to see how they combine.
You already know Faraday's law (a changing B field induces an electric field) and Ampere's law (currents and changing E fields produce magnetic fields). The Lorentz force law F⃗ = q(E⃗ + v⃗ × B⃗) is the complementary statement about how those fields act on matter: a charge q moving at velocity v⃗ through both an electric field E⃗ and a magnetic field B⃗ experiences forces from both simultaneously. The electric piece qE⃗ acts in the direction of the field regardless of how the charge is moving. The magnetic piece q(v⃗ × B⃗) requires motion — a stationary charge feels no magnetic force — and from the cross-product you already know, it acts perpendicular to both the velocity and the magnetic field. This perpendicularity means the magnetic force does no work on the charge; it can bend the trajectory but never speed up or slow down the particle.
Together, Lorentz force plus Maxwell's four equations form the complete framework of classical electromagnetism. The four equations are: Gauss's law for E (∇·E⃗ = ρ/ε₀, charges source field lines), Gauss's law for B (∇·B⃗ = 0, no magnetic monopoles), Faraday's law (∇ × E⃗ = −∂B⃗/∂t, changing B induces E), and the Ampere-Maxwell law (∇ × B⃗ = μ₀J⃗ + μ₀ε₀ ∂E⃗/∂t, currents and changing E produce B). The last term — the displacement current μ₀ε₀ ∂E⃗/∂t — was Maxwell's key addition. Without it, the Ampere-Maxwell law would be inconsistent with charge conservation, and the framework would not support electromagnetic waves.
The real power of Maxwell's equations is what you can derive from them together. Taking the curl of Faraday's law and substituting the Ampere-Maxwell law yields a wave equation for E⃗ with speed c = 1/√(μ₀ε₀). When you plug in the values of those constants, out comes the speed of light — a prediction that electricity and magnetism were not separate phenomena but two aspects of a single electromagnetic field. This is one of the great unifications in physics history: optics, electricity, and magnetism are the same thing viewed from different contexts.
The unification also has a deeper significance that your prerequisites set up. Faraday's law and Ampere's law, which you learned as separate experimental facts, turn out to be intimately linked: a changing E field creates B, and a changing B field creates E. These mutual inductions sustain each other as a self-propagating wave traveling through empty space. The Lorentz force law then closes the loop by describing how that wave, once created, exerts forces on the charged matter it encounters — from radio antennas to the retina of your eye.