In front of uniform light beam of large cross-sectional area and intensity $I$, a solid sphere of radius $R$ which is perfectly reflecting, is placed. Find the force exerted on this sphere due to the light beam.

To find the force on sphere, we consider a small elemental strip of angular width $d\theta$on its surface at an angle θ from its horizontal diameter as shown. The area $dS$ of this strip on the surface of sphere is 

$dS=2\pi R\sin \theta .Rd\theta \dots \dots \left(1\right)$

Now as shown in figure, $dA$ is the projection of the slant strip area $dS$ along the cross-sectional plane of the light beam and it is given as

$dA=dS\cos \theta \dots \dots \left(2\right)$

Here the power of light incident on this strip is $dP=IdS\cos \theta$

Thus the momentum of photons per second incident on this strip are

$\frac{dp}{dt}=\frac{dP}{c}=\frac{IdA}{c} \dots \dots \left(3\right)$

Here we can see that these photons are incident at an angle $\theta$ to the normal N of this strip and as sphere is perfectly reflecting. These are reflected at the same angle $\theta$ to N as shown in figure.

Here the change in momentum of photons is along the normal and thus force exerted on this strip along the normal is

$dF=2\frac{dp}{dt}\cos \theta =\frac{2IdA\cos \theta }{c} \dots \dots \left(4\right)$

Thus net force on sphere will be given as

$$\begin{gathered} F=\int dF\cos \theta \\ =\int \frac{2IdA{\cos }^2\theta }{c} \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{2I\left(2\pi R\sin \theta \cos \theta .Rd\theta \right){\cos }^2\theta }{c} \\ =\frac{4\pi R^{2}I}{c}{\int }_0^{\frac{\mathrm{\pi }}{2}}{\cos }^3\theta \sin \theta d\theta \\ =\frac{4\pi R^{2}I}{c}\left(\frac{1}{4}\right) \\ \Rightarrow F=\frac{\pi R^{2}I}{c} \dots \dots \left(5\right) \end{gathered}$$

If the sphere is non reflecting, from equation (3) we find the total force

$F\text{'}=\int \frac{IdA}{c}=\frac{I}{c}\left({\int }_{\theta =0}^{\theta =\frac{\mathrm{\pi }}{2}}dA\right)=\frac{I\left({\mathrm{\pi R}}^2\right)}{c}=\frac{{\mathrm{\pi R}}^2\mathrm{I}}{c} ... \left(6\right)$

We can see that equation (6) is exactly same as equation (5). 

Thus for a sphere placed in the path of a light beam, force exerted on sphere is independent of the nature of the surface of sphere. But this happens only for sphere.