Question

Consider a plane wave incident on a convex lens of diameter 5cm and of focal length 10cm. If the wavelength of the incident light is 6000°A, calculate the radius of the first dark ring on the focal plane of the lens. Repeat the calculation for a lens of same focal length but diameter 15cm. Interpret the result physically.

Answer 1

When a circular aperture of radius $a$ placed in front of a convex lens of focal length $f$ and illuminated by a parallel beam of light of wavelength $\lambda$ then the radius of the first dark ring is given as follows:

$R=\frac{3.832\lambda f}{2\pi a}$

then, for $a=2.5\, cm$

$R=\frac{3.832\cdot 6\cdot10^{-7}\cdot 0.1}{2\cdot\pi \cdot0.025} = 1.46\cdot 10^{-7} m$

and for $a=7.5\, cm$

$R=\frac{3.832\cdot 6\cdot10^{-7}\cdot 0.1}{2\cdot\pi \cdot0.075} = 4.88\cdot 10^{-8} m$