What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5 D? ('D' stands for dioptre).
Show Hint
- 0.04
- 0.40
- 0.1
- 0.01
The Correct Option is A
Approach Solution - 1
To determine the relative decrease in the focal length of a lens when the optical power changes, we need to understand the relationship between optical power and focal length.
The optical power \( P \) of a lens is given by the formula:
\(P = \frac{1}{f}\)
where:
- \(P\) is the power in dioptres (D)
- \(f\) is the focal length in meters
Initially, the optical power of the lens is \( P_1 = 2.5 \, \text{D} \). Therefore, the focal length \( f_1 \) is:
\(f_1 = \frac{1}{P_1} = \frac{1}{2.5} = 0.4 \, \text{m}\)
The optical power is increased by \( 0.1 \, \text{D} \) to become \( P_2 = 2.6 \, \text{D} \). The new focal length \( f_2 \) is:
\(f_2 = \frac{1}{P_2} = \frac{1}{2.6} \approx 0.3846 \, \text{m}\)
The decrease in focal length is:
\(\Delta f = f_1 - f_2 = 0.4 - 0.3846 = 0.0154 \, \text{m}\)
The relative decrease in focal length is calculated by the formula:
\(\text{Relative decrease} = \frac{\Delta f}{f_1} = \frac{0.0154}{0.4} = 0.0385\)
Rounding to two decimal places, the relative decrease is approximately 0.04.
Therefore, the correct answer is 0.04.
Approach Solution -2
Step 1: Calculate the initial focal length
The relationship between optical power \( P \) and focal length \( f \) is given by:
\[ P = \frac{1}{f}. \] Thus, the initial focal length is: \[ f_1 = \frac{1}{P_1} = \frac{1}{2.5} = 0.4 \, \text{m}. \]
Step 2: Calculate the new focal length
After the optical power increases by 0.1 D, the new optical power becomes:
\[ P_2 = 2.5 + 0.1 = 2.6 \, \text{D}. \] The new focal length is: \[ f_2 = \frac{1}{P_2} = \frac{1}{2.6} \approx 0.3846 \, \text{m}. \]
Step 3: Calculate the relative decrease in focal length
The relative decrease in focal length is: \[ \text{Relative decrease} = \frac{f_1 - f_2}{f_1} = \frac{0.4 - 0.3846}{0.4} = \frac{0.0154}{0.4} \approx 0.04. \]
Final Answer:
The correct answer is option (1) \( \boxed{0.04} \).
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