The box of a pin hole camera of length L has a hole of radius a. It is assumed that when the hole is illuminated by a parallel beam of light of wavelength `lambda` the speed of the spot (obtained on the opposite wall of the camera) is the sum of its geometrical spread and the spread due to diffraction. The spot would then have its minimum size (say `b_("min"))` when .....

Options

A. `alpha=(lambda^(2))/(L) and b_(min)=((2lambda^(2))/(L))`
B. `a=sqrt(lambdaL)and b_(min)=((2lambda^(2))/(L))`
C. `a=sqrt(lambdaL) and b_(min)= sqrt(4lambdaL)`
D. `alpha=(lambda^(2))/(L) and b_(min)= sqrt(4 lambda L)`

Answer

C

Explanation

Condition for first mimimum,
In `d sin theta= n lambda`
`d=a sin theta=(y)/(D)=(a)/(L)`
`n=1`
`:.axx(a)/(L)= lambda`
`:.a^(2)=lambdaL`
`:.a=sqrt(lambdaL)`
Size of spot means diameter `=a+a`
`=2a`
`=2sqrt(lambdaL)`
`=sqrt(4lambdaL)`