Question

The box of a pinhole 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 λ the spread 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:

$straight a equals space straight lambda squared over straight L space and space straight b subscript min space equals space open parentheses fraction numerator 2 straight lambda squared over denominator straight L end fraction close parentheses$
$straight a space equals space square root of λL space and space straight b subscript min space equals open parentheses fraction numerator 2 straight lambda squared over denominator straight L end fraction close parentheses$
$straight a space equals space square root of λL space and space straight b subscript min space equals square root of space open parentheses 4 λL close parentheses end root$
$straight a space equals space straight lambda squared over straight L space and space straight b subscript min space equals space square root of 4 λL end root$

Solution

straight a space equals space square root of λL space and space straight b subscript min space equals square root of space open parentheses 4 λL close parentheses end root

In diffraction, first minima, we have
sin θ = λ/a

So , size of a spot
b= 2a +2Lλ/a (i)
Then, minimum size of a spot, we get
$fraction numerator partial differential straight b over denominator partial differential straight a end fraction space equals space 0 straight lambda rightwards double arrow space 1 minus Lλ over straight a squared space equals space 0 straight a space equals space square root of λL straight b subscript min space equals space 2 square root of λL space plus space 2 square root of λL space space left parenthesis ii right parenthesis space left square bracket by space substtiuting space the space value space ofa space from space eq. space left parenthesis ii right parenthesis space in space eq space left parenthesis straight i right parenthesis equals space 4 space square root of λL So comma space the space radius space of space the space spot comma space straight b subscript min space equals space 4 over 2 square root of λL space equals space square root of 4 λL end root$

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