Question

The wavelength of a certain colour of light in vacuum is $572nm$. What is its wavelength in diamond, which has an index of refraction of 2.42?
A) $236nm$
B) $1384nm$
C) $3350nm$
D) $97.7nm$
E) $574.42nm$

Answer 1

Hint: The speed of light of any colour in a vacuum is the speed of light. When the light enters a medium other than the vacuum, it slows down according to the optical density of the medium. So the wavelength of the light changes when it moves from medium to medium.

Complete step by step answer:
In our problem, it is given that the wavelength of the light in a vacuum is $572nm$. We know that the wavelength of light in a medium is inversely proportional to the refractive index of a medium. So, we can write, $\mu \propto \dfrac{1}{\lambda }$.
Suppose the light given in our problem has a wavelength ${{\lambda }_{c}}$ in vacuum (refractive index ${{\mu }_{c}}=1$) and the wavelength of the light changes into ${{\lambda }_{d}}$ when it goes into the diamond (refractive index ${{\mu }_{d}}=2.42$), so we can write both these conditions as,
${{\mu }_{c}}\propto \dfrac{1}{{{\lambda }_{c}}}$…. Equation (1)
${{\mu }_{d}}\propto \dfrac{1}{{{\lambda }_{d}}}$ ….. equation (2)
Taking the ratio of equation (2) to equation (1), we get,
$\dfrac{{{\mu }_{d}}}{{{\mu }_{c}}}=\dfrac{{{\lambda }_{c}}}{{{\lambda }_{d}}}$
Substituting the values of wavelength and refractive index into the equation above,we get,
$\dfrac{2.42}{1}=\dfrac{572nm}{{{\lambda }_{d}}}$
$\Rightarrow {{\lambda }_{d}}=\dfrac{572nm}{2.42}$
$\therefore {{\lambda }_{d}}=236.36nm\approx 236nm$
So, the answer to the question is option (A) $236nm$

Note: The ratio of speed of light in vacuum to the speed of light in a particular medium is known as the refractive index of the medium. We can express it mathematically as,
$\mu =\dfrac{c}{v}$
Where,
c is the velocity of light in vacuum.
v is the velocity of light in medium.
The only quantity which doesn’t change while the light travels from one medium to another medium is the frequency of the light.
Refractive index of a material is a function of wavelength, which a material shows a different refractive index with lights of a different wavelength. This is the cause of dispersion, which is seen in a prism.