The ration of resolving powers of an optical microscope for two wavelangths
`lamda_(1) =4000` `Å and lamda_(2)=6000 Å is`

Options

A. `8:27`
B. `9:4`
C. `3:2`
D. `16:81`

Answer

To find the ratio of the resolving powers of an optical microscope for two different wavelengths, we can follow these steps:

### Step 1: Understand the relationship between resolving power and wavelength
The resolving power (R) of an optical instrument, such as a microscope, is inversely proportional to the wavelength (λ) of the light used. This can be expressed mathematically as:
\[ R \propto \frac{1}{\lambda} \]

### Step 2: Set up the ratio of resolving powers
Let \( R_1 \) be the resolving power for wavelength \( \lambda_1 \) and \( R_2 \) be the resolving power for wavelength \( \lambda_2 \). According to the relationship established in Step 1, we can write:
\[ \frac{R_1}{R_2} = \frac{\lambda_2}{\lambda_1} \]

### Step 3: Substitute the given values
Given:
- \( \lambda_1 = 4000 \, \text{Å} \)
- \( \lambda_2 = 6000 \, \text{Å} \)

We can substitute these values into the ratio:
\[ \frac{R_1}{R_2} = \frac{6000 \, \text{Å}}{4000 \, \text{Å}} \]

### Step 4: Simplify the ratio
Now, we simplify the fraction:
\[ \frac{R_1}{R_2} = \frac{6000}{4000} = \frac{6}{4} = \frac{3}{2} \]

### Step 5: State the final answer
Thus, the ratio of the resolving powers of the optical microscope for the two wavelengths is:
\[ \frac{R_1}{R_2} = \frac{3}{2} \]

### Final Answer:
The ratio of the resolving powers of the optical microscope for the two wavelengths \( \lambda_1 = 4000 \, \text{Å} \) and \( \lambda_2 = 6000 \, \text{Å} \) is \( \frac{3}{2} \).

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