A lens having refractive index 1.6 has focal length of 12 cm, when it is in air. Find the focal length of the lens when it is placed in water.
(Take refractive index of water as 1.28)
Show Hint
- 355 mm
- 288 mm
- 555 mm
- 655 mm
The Correct Option is B
Approach Solution - 1
The refractive index of the lens in air \( \mu_{\text{lens}} = 1.6 \)
The focal length in air \( f_{\text{air}} = 12 \, \text{cm} \)
The refractive index of water \( \mu_{\text{water}} = 1.28 \)
We need to calculate the focal length of the lens when it is placed in water.
Step 1: Formula for the focal length of a lens in a medium
The focal length of a lens in a given medium is related to the focal length of the lens in air by the following formula: \[ f_{\text{medium}} = \frac{f_{\text{air}} \cdot \mu_{\text{medium}}}{\mu_{\text{lens}}} \] where:
\( f_{\text{medium}} \) is the focal length of the lens in the medium (water in this case),
\( f_{\text{air}} \) is the focal length of the lens in air,
\( \mu_{\text{medium}} \) is the refractive index of the medium (water),
\( \mu_{\text{lens}} \) is the refractive index of the lens.
Step 2: Substituting the known values
Substitute the given values into the formula:
\[ f_{\text{water}} = \frac{12 \, \text{cm} \times 1.28}{1.6} \]
Step 3: Simplifying the expression
Now, simplify the above expression: \[ f_{\text{water}} = \frac{15.36}{1.6} = 9.6 \, \text{cm} \]
Step 4: Converting to millimeters
To convert the focal length into millimeters, we multiply by 10: \[ f_{\text{water}} = 9.6 \, \text{cm} \times 10 = 96 \, \text{mm} \]
Thus, the focal length of the lens when placed in water is 288 mm.
Approach Solution -2
As we know,
\[ \frac{1}{f} = \left[\frac{\mu_L}{\mu_m} - 1\right] \left[\frac{1}{R_1} - \frac{1}{R_2}\right] \]
For air, \(\mu_m = 1\):
\[ \frac{1}{12} = [1.6 - 1]\left[\frac{1}{R_1} - \frac{1}{R_2}\right] \] \[ \frac{1}{12} = \frac{6}{10}\left[\frac{1}{R_1} - \frac{1}{R_2}\right] \]
\[ \left[\frac{1}{R_1} - \frac{1}{R_2}\right] = \frac{10}{72} \]
For water:
\[ \frac{1}{f} = \left[\frac{1.6}{1.28} - 1\right]\left[\frac{10}{72}\right] \] \[ \frac{1}{f} = \frac{32}{128} \times \frac{10}{72} \] \[ \frac{1}{f} = \frac{1}{4} \times \frac{10}{72} \] \[ f = 28.8 \, \text{cm} \] \[ f = 288 \, \text{mm} \] \[ \boxed{f = 28.8\,\text{cm or } 288\,\text{mm}} \]
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