# a) A film of magnesium fluoride, (n=1.38), 1.25 x 10^{-5} cm thick, is used to coat a camera lens...

## Question:

a) A film of magnesium fluoride, (n=1.38), 1.25 x {eq}10^{-5} {/eq} cm thick, is used to coat a camera lens (n=1.55).

Are any wavelengths in the visible spectrum intensified in the reflected light?

b) Bats use the reflections from ultra-high frequency sound to locate their prey.

Estimate the typical frequency of a bat's sonar. Take the speed of sound to be 3.40 x {eq}10^2 {/eq} m/s, and a small moth 3.00 mm across, to be a typical target.

## Interference of Waves:

Two waves of the same frequency and that are coherent will form a stationary pattern of maxima and minima that correspond to two waves being in phase or out of phase. This phenomenon is called an interference of waves.

a) The condition for the light to interfere constructively upon reflection can be written as follows:

{eq}2 d n = m \lambda {/eq}

Here

• {eq}d = 1.25\times 10^{-7} \ m {/eq} is the thickness of the film;
• {eq}n = 1.38 {/eq} is the index of refraction of the coating film;

The wavelengths that will be intensified in reflection are given by:

{eq}\lambda = \dfrac {2dn}{m} {/eq}

Substituting the numbers, we get:

{eq}\lambda = \dfrac {2\times 1.25\times 10^{-7} \ m \times 1.38}{m} = \dfrac {3.45 \times 10^{-7} \ m}{m} {/eq}

It is clear that the largest wavelength intensified in reflection is

{eq}\lambda_{max} = \dfrac {3.45\times 10^{-7} \ m}{1} = 3.45\times 10^{-7} \ m {/eq}

This wavelength corresponds to the ultraviolet light, so no wavelengths in the visible spectrum will be intensified in reflection.

b) The sound wave will effectively reflect from the target if the target is larger than the wavelength of the wave. Estimate for the frequency of a bat's sound wave can be made as follows:

{eq}f = \dfrac {c}{\lambda} = \dfrac {c}{d} {/eq}

Here

• {eq}c = 340 \ m/s {/eq} is the speed of sound in the air;
• {eq}d = 3 \ mm {/eq} is the size of the prey for the bat;

Evaluating, we get:

{eq}f = \dfrac {340 \ m/s}{3\times 10^{-3} \ m} \approx \boxed{1.13\times 10^5 \ Hz} {/eq}