I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting aside the physical part of the question, I needed to do the following:-
Find the net momentum per unit area of the beam $I/c$ and multiply this with a small area on my sphere to ultimately integrate over half a sphere to get the net momentum.
My efforts
Consider a sphere of $radius=r$ placed with the center on origin of the co-ordinate axes. Let the position vector characterizing the sphere be $$\vec r=r \cos \theta \hat z+r \sin \theta \cos \phi \hat x+ r \sin \theta \sin \phi \hat y$$ Where $\theta$ is the angle of the position vector with $z$ axis and $\phi$ the angle of the projection of the position vector in x-y plane with x axis.
Now, suppose the beam is falling along the $x$axis. Therefore at a point $(x,y.z)$ the incoming beam will make an angle $\phi$ with the normal at that point. Therefore, for an area $da=r^2 \sin \theta d\theta d\phi$, the momentum imparted will be $\frac{I}{c} da \cos \phi (-\hat x)$, since the area of the beam imparting momentum on the area $da$ is only $da \cos \phi$ normal to the beam. Therefore the net momentum should be $$\int_0^\pi \int_{-\frac \pi 2}^{\frac \pi 2} \frac {r^2I}c \cos \phi \sin \theta d\phi d\theta (-\hat x)$$
But that evaluates to $\frac {4r^2I}c$. But the answer and intuitive understanding ($\pi r^2$ area of the beam is intercepted and absorbed so the net momentum is this area times the momentum per unit area) both gives $\pi r^2I/c$ as the answer. Where did I go wrong?
The same problem arises if we consider a reflecting sphere. I guess the problem is in dealing with the projected areas, i.e the projection of $da$ as $da \cos \phi$ normal to the direction of the incoming beam.
Sorry if this has a trivial mistake. Can anyone help me with solving the perfectly reflecting case properly? I do not know how to proceed therein appropriately( in reflection, the laws of reflection need to be followed and the given answer is the same as calculated for the absorption case)