How Is Poynting Flux Calculated on a Sphere with Angled Incidence?

[bdforbes]

Problem: what is the Poynting flux on a sphere if the Poynting vector S makes an angle of theta with the normal?

My instinct is S*cos(theta), but the overall problem I'm looking at implies it is S*cos(theta)^2. Can anyone help me on this?

 

[gabbagabbahey]

$$\text{Flux}= \int_{\mathcal{S}} \vec{S} \cdot \vec{da}$$

Does that help?

 

[bdforbes]

Well yes, that implies I should be using S*cos(theta). But that just doesn't fit with the solutions I have. Maybe I should give the full problem.

Calculate the radiation force on a sphere in terms of the Poynting vector.

The solution calculates the force on a thin ribbon; I've attached the diagram.

The second term in brackets is the area of the strip, the sin(theta) at the end is to pick out only the force to the right (since the other components cancel by symmetry), so the first term must be the radiation pressure. The radiation pressure for light normally incident on a reflecting surface is 2S/c. For off-normal incidence, I would include a factor of cos(phi), but the solutions have cos(phi)^2. What is going on here?

 

[bdforbes]

I've figured out this problem. The issue is that we are dealing with an incident energy volume density, and we also have to account for off-normal incidence. The differential volume contributing to the pressure at a point on the sphere varies as cos(phi), and the pressure is only due to normally incident force, so we pick up another factor of cos(phi). Essentially we are accounting for reduced cross-sectional area, and disregarding shear forces.