Tutorial: Maxwell stress tensor and radiation pressure

Light is obliquely incident on a perfectly absorbing screen at an angle \( \theta \) relative to the surface normal. Steps 1-7 should be carried out for two different polarizations of the wave: (a) \( \mathbf{B} \) field parallel to the surface, and (b) \( \mathbf{E} \) field parallel to the surface. Then proceed to step 8.

  1. Choose axis \( z \) normal to the surface and another axis parallel to (a) \( \mathbf{B} \) or (b) \( \mathbf{E} \).
  2. Write down the Maxwell stress tensor as a matrix (all components).
  3. Which component gives the radiation pressure?
  4. Express the pressure in terms of the energy density of the radiation.
  5. Is there a momentum transfer parallel to the surface? Find it or argue why it is absent.
  6. Write down the full Maxwell stress tensor in the reference frame where the axis \( z' \) is parallel to the wavevector of the incident wave.
  7. Rationalize the result of the previous step.
  8. What if the incident light is unpolarized?
  9. Use general properties of tensors to find the radiation pressure from the result of item #6. (Use a rotation matrix.) Compare with the result of item #4 (they should match).
  10. Give a physical interpretation of the angular dependence of the radiation pressure.
  11. How do the above results (both pressure and transverse momentum transfer) change if the surface is perfectly reflecting instead of absorbing?
  12. A plane wave is incident on a perfectly reflecting sphere of radius \( R \). Find the total radiation force acting on the sphere. (You will need to set up an integral over the illuminated half of the sphere.)
  13. Answer the same question for a perfectly absorbing sphere.