Wavelength of light used in an optical instrument are `lambda_(1) = 4000 Å` and `lambda_(2) = 5000Å` then ratio of their respective resolving powers (corresponding to `lambda_(1)` and `lambda_(2)`) is

Options

A. `16:25`
B. `9:1`
C. `4:5`
D. `5:4`

Explanation

To find the ratio of the resolving powers corresponding to two different wavelengths of light used in an optical instrument, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Resolving Power**:
Resolving power (R) of an optical instrument is defined as its ability to distinguish between two closely spaced objects. It is inversely proportional to the wavelength of the light used. This means that as the wavelength decreases, the resolving power increases.

2. **Write the Relationship**:
The relationship between resolving power and wavelength can be expressed as:
\[ R \propto \frac{1}{\lambda} \]
Therefore, for two wavelengths \( \lambda_1 \) and \( \lambda_2 \), the ratio of their resolving powers can be written as:
\[ \frac{R_1}{R_2} = \frac{\lambda_2}{\lambda_1} \]

3. **Substitute the Given Values**:
We are given:
- \( \lambda_1 = 4000 \, \text{Å} \)
- \( \lambda_2 = 5000 \, \text{Å} \)

Now substituting these values into the equation:
\[ \frac{R_1}{R_2} = \frac{5000 \, \text{Å}}{4000 \, \text{Å}} \]

4. **Simplify the Ratio**:
Simplifying the fraction:
\[ \frac{R_1}{R_2} = \frac{5000}{4000} = \frac{5}{4} \]

5. **Final Result**:
Therefore, the ratio of the resolving powers corresponding to \( \lambda_1 \) and \( \lambda_2 \) is:
\[ R_1 : R_2 = 5 : 4 \]

### Conclusion:
The ratio of the respective resolving powers is \( 5 : 4 \).