Questions: Momentum Density in Electromagnetic Fields

Question 1

A laser beam strikes an absorbing surface and exerts a measurable force on it. According to the concept of EM field momentum, what is the physical origin of this force?

• A. The electric field directly repels free electrons in the surface outward
• B. The magnetic Lorentz force on charges set in motion by the wave's electric field pushes the surface forward along the propagation direction
• C. Radiation pressure is purely a quantum effect with no classical explanation
• D. The oscillating electric field creates eddy currents whose heating expands the surface

Answer: (B)

The mechanism is classical and purely Lorentz-force based. The wave's electric field accelerates charges in the surface. Those moving charges then experience a magnetic Lorentz force from the wave's B field, directed along the propagation direction. This forward push is the transfer of field momentum to matter. Option C is wrong: radiation pressure is a classical result, fully derivable from Maxwell's equations and confirmed quantum mechanically later.

Question 2

Compared to a perfect absorber, the radiation pressure on a perfect reflector for the same incident wave is:

• A. The same — the wave's energy is unchanged upon reflection, so the pressure is unchanged
• B. Half as much — only the incoming wave contributes, not the reflected one
• C. Twice as much — the field momentum reverses direction, doubling the impulse delivered to the surface
• D. Four times as much — both the electric and magnetic field momenta must be counted separately

Answer: (C)

Force equals rate of momentum transfer. For an absorber, the incoming field momentum is fully transferred to the surface. For a reflector, the incoming momentum is transferred AND the reflected wave carries momentum in the opposite direction — so the surface provides that momentum too. The total change in field momentum is 2|g| per unit time, giving twice the force. The same principle applies in mechanics: catching a ball and throwing it back imparts twice the impulse of just catching it.

Question 3

Momentum in electromagnetic fields is a quantum mechanical effect that has no place in classical electrodynamics.

• T. True
• F. False

Answer: False

EM field momentum is a purely classical result, derivable from Maxwell's equations and Newton's laws of momentum conservation. The momentum density g = ε₀(E × B) = S/c² follows from requiring that momentum be conserved when light interacts with charged matter. Quantum mechanics later confirmed this result at the photon level (p = E/c for massless photons), but the classical field theory already anticipates it. The two are consistent precisely because the photon picture is the quantization of the classical field.

Question 4

The ratio of momentum density to energy density in an electromagnetic wave equals 1/c.

• T. True
• F. False

Answer: True

For a plane wave, energy density u = ε₀E² (SI units) and Poynting vector S = cu (since the wave travels at c). Momentum density g = S/c² = u/c. So g/u = 1/c. This is exactly the massless-particle relation p = E/c, which for photons gives p = hf/c = h/λ. The classical field result and the quantum photon result are consistent — field momentum per unit volume equals the quantum-mechanical count of photon momenta.

Question 5

Why does a perfect reflector experience twice the radiation pressure of a perfect absorber, even though it absorbs no energy from the incident wave?

Model answer: Radiation pressure results from the transfer of field momentum to the surface. For a perfect absorber, all the incoming field momentum (directed forward) is transferred to the surface. For a perfect reflector, the surface must supply the incoming forward momentum to the wave upon reflection AND reverse its direction — effectively transferring 2|g| of momentum to itself. The force on the surface is the rate of momentum transfer, which is doubled for a reflector.

The key is that force depends on the change in momentum of the field, not the change in energy. A perfect reflector does zero net work on the wave (the wave's energy is conserved), but it completely reverses the wave's momentum. By Newton's third law, the surface receives a force equal to twice the incoming momentum flux. An analogy: catching a rubber ball and throwing it back imparts twice the impulse of catching it and holding it still.