The box of a pin hole camera, of length L, has a hole of radius a. It… | JEE Main 2016

Q78JEE Main 2016Difficult

Diffraction at a Single Slit

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 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:

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Answer: (a)

Spot size = geometric spread + diffraction spread: $b = a + \dfrac{\lambda L}{a}$
Minimum: $\dfrac{db}{da} = 1 - \dfrac{\lambda L}{a^2} = 0 \Rightarrow a = \sqrt{\lambda L}$
$b_{\min} = \sqrt{\lambda L} + \sqrt{\lambda L} = 2\sqrt{\lambda L} = \sqrt{4\lambda L}$
So the answer is (a).

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