# Absorption/Reflection coefficient independency of radiation pressure exerted on sphere

Prove that the force exerted on a sphere of radius $$r$$ by a light source of intensity $$I$$ is not affected even if the sphere is not perfectly absorbing.

Question 11,Photoelectric effect and wave-particle duality, Concepts of physics Volume-2 by Dr.HC Verma

The derivation for the force exerted on a perfectly reflecting sphere can be found here

A slight manipulation of eq-2.58 yields:

$$dF=dp(2R+a)\cos\theta=\frac{(2R+a)IdA}{c}\cos\theta$$

where $$R$$ and $$a$$ are the reflection and absorption coefficients of the sphere respectively.

Thus the final result would be:

$$F=\frac{I\pi r^2(2R+a)}{2c}$$

Its clear that for $$R=1$$ and $$a=0$$, the result is as expected for a perfectly reflecting sphere.

However, coming back to the initial question, the final expression clearly varies for different values of $$R$$.

The momentum/force, obviously varies for different absorbing coefficients of the sphere. This is contradicting the question. Where have I gone wrong? Or is the question incorrect?

• So now let us consider 2 cases- one perfectly reflecting and the other perfectly absorbing. In the first case, considering the force to be $F=\int dF\cos\theta=2\int\frac{dp}{dt}\cos\theta$ while the second one must be $F=\int dF=\int\frac{dp}{dt}$(since it is already along the path of light). Am I right in saying this? If yes, the second integral gives us the force $F=\frac{2I\pi R^2}{c}\int_0^{\pi/2}{cos^2\theta\sin\theta d\theta}=\frac{2I\pi R^2}{3c}$ – newbie105 Nov 20 '20 at 16:13
• Yes, but the final $F$ should be $F$ = $\frac{2IπR^2}{c} \int_0^{π/2} cosθsinθ \,dθ$ , Notice there is another cosine term gone, in the perfectly reflecting case $dF =2 dp (cosθ)$ - Eq 2.58, while in the perfectly absorbing case $dF = dp$ In the end it should give you $F = \frac{IπR^2}{c}$ In the perfectly reflecting case, one cosine was for taking the component of momentum along the normal to the sphere, the other was for taking the component of that horizontally (pointing to the right). In the perfectly absorbing case, these two are not needed, but it needs to be divided by 2. – bluemystic Nov 20 '20 at 16:56