Question

A beam of monochromatic light with a wavelength of 400 nm in air travels into water. what is the wavelength of the light in water? give your answer in nanometers.

Answer 1

The refractive index of water is $n=1.33$. This means that the speed of the light in the water is:
$v= \frac{c}{n}= \frac{3 \cdot 10^8 m/s}{1.33 }=2.26 \cdot 10^8 m/s$

The relationship between frequency f and wavelength $\lambda$ of a wave is given by:
$\lambda= \frac{v}{f}$
where v is the speed of the wave in the medium. The frequency of the light does not change when it moves from one medium to the other one, so we can compute the ratio between the wavelength of the light in water $\lambda_w$ to that in air $\lambda$ as
$\frac{\lambda_w}{\lambda}= \frac{ \frac{v}{f} }{ \frac{c}{f} } = \frac{v}{c}$
where v is the speed of light in water and c is the speed of light in air. Re-arranging this formula and by using $\lambda=400 nm$, we find
$\lambda_w = \lambda \frac{v}{c}=(400 nm) \frac{2.26 \cdot 10^8 m/s}{3 \cdot 10^8 m/s}=301 nm$
which is the wavelength of light in water.