Question

The wavelength of light while it is passing through water is \(540\,\text{nm}\). The refractive index of water is \( \frac{4}{3} \). The wavelength of the same light when it is passing through a transparent medium having refractive index of \( \frac{3}{2} \) is _________ nm.

• A. \(480\)
• B. \(840\)
• C. \(380\)
• D. \(540\)

Hint:

Frequency of light remains unchanged when it passes from one medium to another; only wavelength changes.

Answer 1

The Correct Option is A

To find the wavelength of light in a transparent medium with a different refractive index, we must first understand how the speed of light and wavelength are affected by the refractive index of the medium. 

The wavelength of light in a medium is related to its wavelength in a vacuum (or air, approximately) by the refractive index of the medium. The formula for this is:

\[\lambda_m = \frac{\lambda_0}{n}\]

where:

• \(\lambda_m\) is the wavelength in the medium,
• \(\lambda_0\) is the wavelength in a vacuum or air,
• \(n\) is the refractive index of the medium.

According to the given data:

• The wavelength of light in water, \(\lambda_{\text{water}}\), is \(540\,\text{nm}\).
• The refractive index of water, \(n_{\text{water}}\), is \(\frac{4}{3}\).
• The refractive index of the new medium, \(n_{\text{medium}}\), is \(\frac{3}{2}\).

The wavelength in water is given by:

\[\lambda_{\text{water}} = \frac{\lambda_0}{n_{\text{water}}}\]

Rearranging to solve for the wavelength in a vacuum, \(\lambda_0\):

\[\lambda_0 = \lambda_{\text{water}} \times n_{\text{water}} = 540\,\text{nm} \times \frac{4}{3}\]

Calculating the above expression:

\[\lambda_0 = 540 \times \frac{4}{3} = 720\,\text{nm}\]

The wavelength in the new medium, \(\lambda_{\text{medium}}\), is then:

\[\lambda_{\text{medium}} = \frac{\lambda_0}{n_{\text{medium}}} = \frac{720\,\text{nm}}{\frac{3}{2}}\]

Solving this gives:

\[\lambda_{\text{medium}} = 720 \times \frac{2}{3} = 480\,\text{nm}\]

Therefore, the wavelength of the light when it passes through the medium with a refractive index of \(\frac{3}{2}\) is \(480\,\text{nm}\).

The correct answer is \(480\).