The photocurrent in the detector assuming all rays to be paraxial w.r.t. lens

Question

A disc shaped photoelectric detector of area 0.5 $cm^2$ generates current when light of frequency > that of yellow light falls on it. The photoelectron generation efficiency is 20%. In the arrangement shown, S is an isotropic 100 watt point monochromatic light source with $\lambda$ =4000 A. The focal of the lens is 20 cm and its circular area is 4 $cm^2$. Find the photocurrent in the detector assuming all rays to be paraxial w.r.t. lens.

My try :

Answer 1

The mistake in your calculation is that you are assuming that the light which is focussed by the lens into a point $S'$ at $60cm$ to the right of the lens acts like a point source which radiates equally in all directions.

This assumption is not correct. All of the light from the lens converges onto $S'$ in a cone, and diverges again only to the right into another cone with the same angle. In the paraxial approximation we can assume this light is spread uniformly across the base of the cone.

When the cone of light which diverges from $S'$ reaches the detector its radius is half that of the aperture of the lens. This can be found from the geometry of similar triangles. So the area $A$ of the circular base of this cone at the detector is $\frac14$ of that of the lens, ie $A=1cm^2$.

The area of the detector is only $0.5cm^2$ and is completely covered by the cone. So only half the light which is collected by the lens subsequently falls onto the detector.