A convex lens of focal length 20 cm is immersed in a liquid of refractive index 1.3. If the refractive index of the material of the lens is 1.5, then the focal length of the lens when immersed in the liquid is:
Show Hint
- \( 20 \) cm
- \( 35 \) cm
- \( \mathbf{65} \) cm
\( 40 \) cm
The Correct Option is C
Solution and Explanation
Step 1: Lens Maker's Formula in a Medium The focal length of a lens in a medium is given by the modified lens maker’s formula: \[ \frac{1}{f_m} = \left( \frac{n_{\text{lens}}}{n_{\text{medium}}} - 1 \right) \frac{1}{f} \] where: - \( f_m \) is the focal length in the medium, - \( n_{\text{lens}} = 1.5 \) is the refractive index of the lens, - \( n_{\text{medium}} = 1.3 \) is the refractive index of the medium, - \( f = 20 \) cm is the original focal length in air.
Step 2: Substituting Given Values \[ \frac{1}{f_m} = \left( \frac{1.5}{1.3} - 1 \right) \frac{1}{20} \] \[ = \left( \frac{1.5 - 1.3}{1.3} \right) \frac{1}{20} \] \[ = \left( \frac{0.2}{1.3} \right) \frac{1}{20} \]
Step 3: Solving for \( f_m \) \[ f_m = \frac{20 \times 1.3}{0.2} \] \[ f_m = \frac{26}{0.2} = 65 \text{ cm} \]
Step 4: Verifying the Correct Option Comparing with the given options, the correct answer is: \[ \mathbf{65} \text{ cm} \]
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