Question

The wavelength of monochromatic light is 5000$A^0$ in air what will be its wave number in a medium of refractive index 1.5?

A. $3\times {10}^6{m}^{-1}$
B. $5\times {10}^6{m}^{-1}$
C. $2\times {10}^6{m}^{-1}$
D. $9\times {10}^6{m}^{-1}$

Answer 1

Hint: When a light of only one color(monochromatic light) is passed into prism that light suffers deviation. We can relate that prism as a sudden medium changing instrument and same thing we can apply in this medium changing case too.
Formula used:
$\begin{array}{rl}& V=v\lambda \\ & {\displaystyle \frac{V_a}{V_m}}={\displaystyle \frac{{\lambda }_a}{{\lambda }_m}}=\mu \end{array}$

Complete answer:
When a monochromatic light passing through one medium gets through the another medium then its frequency will remain constant which makes its speed and wavelength to change.
Velocity of light is given as $V=v\lambda$ where V is the velocity and v is the frequency and $\lambda$is the particular medium wavelength.
We have from snell’s law this relation when light travels from air(a) to the other medium(m) of refractive index $\mu$which is
${\displaystyle \frac{V_a}{V_m}}={\displaystyle \frac{{\lambda }_a}{{\lambda }_m}}=\mu$
According to the question given wavelength of the light in air is 5000 angstroms and refractive index of the medium is 1.5. by substituting these values in the above equation we get
$\begin{array}{rl}& {\lambda }_m={\displaystyle \frac{{\lambda }_a}{\mu }}\\ & {\displaystyle \frac{1}{{\lambda }_m}}={\displaystyle \frac{\mu }{{\lambda }_a}}\end{array}$
We were asked to find out the wave number which will be the inverse of the wavelength.
So we have ${\displaystyle \frac{1}{{\lambda }_m}}={\displaystyle \frac{\mu }{{\lambda }_a}}={\displaystyle \frac{1.5}{5\times {10}^{-7}}}=3\times {10}^6{m}^{-1}$

So, the correct answer is “Option A”.

Additional Information:
White light is an EM wave which is made up of various colored rays with different wavelengths. Now when this light wave is passed into any transparent material time of interaction of individual constituent rays with atoms of that material determines the structure of the light coming out the prism.

Note:
The reason why velocity changes when it enters a different medium is because of variation in the optical density. As optical density changes, velocity of light changes which in turn changes its wavelength too. And frequency will be constant as the source is the same and not moving. We can see the doppler effect in light also which is slightly different from sound doppler effect.