There is a parallel beam of light of larger aperture . Let I be the intensity of light . Perfectly reflecting sphere of radius r is kept in the path of this parallel beam . Find the force exerted by the light beam on sphere .

Explanation

Let us select a reference line OA opposite to the direction of incident beam. Here O is centre of the sphere .

Let us select a radius OP of sphere at an angle `theta` with the reference line OA . Rotate this radius OP around OA to get a circle on sphere . Now increase this angle to `theta+dtetha` and again rotate OP about OA to get one more circle. Radius of this circular ring segment is `r sin tetha` .
Total are of this ring segment will be `2pi (r sin theta) ("rd"theta) = 2pir ^2 sin theta d theta.`
It we select some are `DeltaA` , then corresponding linear momentum falling on the same surface will be `DeltaU//c` . Light will be reflected at the smae angle with the normal and hence change in linear momentum can be written as follows :
`Deltap//Deltat=(2//c)I DeltaA cos^2theta`
The above expression is basically for the force acting on `DeltaA` portion of ring segment . This force acts perpendicular to the surface of the ring. And from symmetry , we can understand that components perpendicular to OA will be canalled and these along OA will be added, hence contribution of this force to the total force will be just component perpendicular to beam and can be written as follows :
`(Deltap//Deltat)costheta=(2//c)IDeltaA cos^3theta`
Now we can replace `DeltaA` with the total area of ring segment to get the force exerted on selected ring segment.
`dF=(2//c)I(2pir^2sintheta"d"theta)cos^3theta`
We can calculate total force on sphere by integrating the above relation for `theta = 0^@ " to" 90^@` because light is falling on the front portion of the sphere.
`F=(4pir^2I)/cint_0^(pi//2)cos^2theta(sintheta"d"theta)`
`F=(4pir^2I)/cint_0^(pi//2)cos^2theta(sintheta"d"theta)" "...(i)`
Let `t =costhetaimpliesdt=-sin theta"d"theta`
Substituting above values in equation (i) , we get the following :
`F=(4pir^2I)/c[(t^3)/4]_1^0`
`impliesF=(pir^2I)/c`