Question

If the refractive index from air to glass is $\frac{3}{2}$ and that from air to water is $\frac{4}{3}$, then the ratio of focal lengths of a glass lens in water and in air is

• A. $1: 2$
• B. $2: 1$
• C. $1: 4$
• D. $4: 1$

Answer 1

The Correct Option is D

To find the ratio of focal lengths of a glass lens when placed in water and in air, we start by understanding the lens maker's formula. The formula for thin lenses is given by:

\(\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\) 

Where:

• \(f\) is the focal length of the lens.
• \(n\) is the refractive index of the material of the lens with respect to the surrounding medium.
• \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.

1. **In Air:**
If the lens is in air, the refractive index of glass with respect to air (\(n_{\text{g/a}}\)) is \(\frac{3}{2}\).

The formula becomes:

\(\frac{1}{f_\text{air}} = \left(\frac{3}{2} - 1\right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) = \frac{1}{2}\left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)

2. **In Water:**
If the lens is in water, first find the refractive index of glass with respect to water (\(n_{\text{g/w}}\)). This can be calculated using relative refractive indices:

\(n_{\text{g/w}} = \frac{n_{\text{g/a}}}{n_{\text{w/a}}} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{9}{8}\)

The lens maker's formula for a lens in water becomes:

\(\frac{1}{f_\text{water}} = \left( \frac{9}{8} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) = \frac{1}{8} \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)

3. **Finding the Ratio of Focal Lengths:**
Given that the same radii are used, we can compare the two equations:

• \(\frac{1}{f_\text{air}} = \frac{1}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)
• \(\frac{1}{f_\text{water}} = \frac{1}{8} \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)

From these equations, the ratio of focal lengths is calculated as:

\(\frac{f_\text{water}}{f_\text{air}} = \frac{\frac{1}{2}}{\frac{1}{8}} = 4\)

Therefore, the ratio of the focal lengths of a glass lens in water to air is \(4:1\).

The correct answer is: \(4:1\).