Momentum Density in Electromagnetic Fields

Core Idea

Electromagnetic fields carry momentum density g = S/c² = (ε₀/c²)(E × B), where S is the Poynting vector. This momentum transfers to matter in the form of radiation pressure, with magnitude equal to energy density divided by c².

Explainer

You've already established from the Poynting vector that electromagnetic fields carry energy, with energy flux S = (1/μ₀)(E × B) measured in watts per square meter. The deeper and perhaps more surprising result is that fields also carry momentum. This isn't obvious classically — momentum seems like a property of matter — but it follows inescapably from the requirement that momentum be conserved when light interacts with matter.

Here's the argument: when an electromagnetic wave hits an absorbing surface and the charges in the surface begin to move, the Lorentz force F = q(E + v×B) has two parts. The electric field accelerates the charges, and then the magnetic field exerts a force on those moving charges. This secondary magnetic force is directed along the propagation direction of the wave — it pushes the surface forward. Something must be carrying that momentum before the wave is absorbed, and that something is the field itself. The momentum density is g = S/c² = ε₀(E × B), pointing in the same direction as the energy flow, with magnitude equal to the energy density divided by c.

The transfer of field momentum to matter is called radiation pressure. For a plane wave with energy density u, the radiation pressure on a perfect absorber is P = u (in SI units of N/m²), and on a perfect reflector it's P = 2u (because the momentum reverses). These are tiny forces in everyday life — the radiation pressure of sunlight on Earth is about 5 μPa — but they become significant in astrophysics (stellar winds blow material away from stars), in optical trapping (laser tweezers grip microscopic beads), and in proposed solar sail spacecraft.

The relationship g = S/c² has a profound implication: energy and momentum in electromagnetic fields are not independent, but tied by g = u/c = energy/(c·volume). This is exactly the relationship for massless particles, and it foreshadows the photon picture in quantum mechanics — photons carry energy E = hf and momentum p = h/λ = E/c. The classical result for field momentum per unit volume matches the quantum mechanical count of photon momenta, confirming that even the classical field theory anticipates the quantum nature of light.