Math 2.0

Optics, electromagnetic theory Maxwell's equations Problem

Abraham–Minkowski controversy

Builds on Maxwell's equationsElectric displacement fieldMagnetic fieldElectric fieldRefractive index

Problem

Let a linear, homogeneous and non-dispersive dielectric medium at rest have permittivity ε\varepsilon and permeability μ\mu. Let EE, BB be the electric and magnetic field, D=εED = \varepsilon E the electric displacement field and H=B/μH = B / \mu. The problem is to decide which density is the momentum density of the electromagnetic field in the medium: gM=D×B(Minkowski),gA=1c2 E×H(Abraham).g_{\mathrm{M}} = D \times B \quad \text{(Minkowski)}, \qquad g_{\mathrm{A}} = \frac{1}{c^2} \, E \times H \quad \text{(Abraham)}. For a light pulse of energy ℏω\hbar \omega in a medium of refractive index nn, the two choices give the momenta nℏω/cn \hbar \omega / c and ℏω/(nc)\hbar \omega / (n c).

Remarks

The two densities differ by the factor n2=εμ/(ε0μ0)n^2 = \varepsilon \mu / (\varepsilon_0 \mu_0), so they agree only in vacuum. Both come from energy–momentum tensors of the field in matter that are consistent with Maxwell's equations in the medium; the Minkowski tensor is not symmetric, the Abraham tensor is. The difference is a momentum carried by the medium, so the question is how the total momentum of the light and the medium is split. A widely accepted view is that both answers are correct for different notions. The Abraham momentum is the kinetic momentum of the light, and the Minkowski momentum is its canonical momentum. Experiments measure the total momentum transfer, and different experiments are explained by one or the other expression once the force on the medium is included.

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  1. Electric displacement field concept
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  3. Magnetic field concept
  4. Maxwell's equations concept
  5. Refractive index concept
  6. Abraham–Minkowski controversy problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Abraham%E2%80%93Minkowski_controversy
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AboutElectric field, Magnetic field
UsesMaxwell's equations, Electric displacement field, Magnetic field, Electric field, Refractive index
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Changed3 November 2026

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