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EEJ MAIN Physics QUESTION #7047
Question 1
A pinhole camera box of length $L$ has a hole of radius $a$. The spot size is the sum of geometrical spread and diffraction spread. At what value of $a$ is the spot size $b_{min}$ minimized?
  • $a = \sqrt{\lambda L}$ and $b_{min} = \frac{2\lambda^2}{L}$
  • $a = \sqrt{\lambda L}$ and $b_{min} = \sqrt{4\lambda L}$✔️
  • $a = \sqrt{\frac{\lambda L}{2}}$ and $b_{min} = 2\sqrt{2\lambda L}$
  • $a = \frac{\lambda^2}{L}$ and $b_{min} = \frac{2\lambda^2}{L}$
Correct Answer Explanation

Spot size $b \approx \text{geometric spread} + \text{diffraction spread} = a + \frac{L\lambda}{a}$.

To minimize $b$, differentiate with respect to $a$ and set to zero:

$\frac{db}{da} = 1 - \frac{L\lambda}{a^2} = 0 \implies a = \sqrt{\lambda L}$.

Substituting back: $b_{min} = \sqrt{\lambda L} + \frac{L\lambda}{\sqrt{\lambda L}} = 2\sqrt{\lambda L} = \sqrt{4\lambda L}$.