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 $\lambda$, 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

• A. $a=\frac{{\lambda }^2}{L}$ and $b_{min}=\frac{{2\lambda }^2}{L}$
• B. $a=\sqrt{\lambda L}$ and $b_{min}=\frac{{2\lambda }^2}{L}$
• C. $a=\sqrt{\lambda L}$ and $b_{min}=\sqrt{4\lambda L}$
• D. $a=\frac{{\lambda }^2}{L}$ and $b_{min}=\sqrt{4\lambda L}$

Answer 1

The correct option is C $a=\sqrt{\lambda L}$ and $b_{min}=\sqrt{4\lambda L}$

We know that spot size $B=2\left(\frac{\lambda L}{2a}\right)+2a$.----(1)
$\frac{\partial B}{\partial a}=0$----(2)

$1-\frac{L\lambda }{a^2}=0⟹a=\sqrt{\lambda L}{B}_{min}=2\sqrt{\lambda L}+2\sqrt{\lambda L}$
by substituting for a from(2) in (1) = $4\sqrt{\lambda L}$
$\therefore \text{The radius of the spot}=0.5\times 4\sqrt{\lambda L}=\sqrt{4\lambda L}$