Light of wavelength `5,000 Å` is falling on a microscope objective lens with a cone angle of `60^@`. Calculate the resolving power of the microscope.

Answer

To calculate the resolving power of a microscope given the wavelength of light and the cone angle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Wavelength of light, \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \)
- Cone angle, \( \theta = 60^\circ \)
- Semi-vertical angle, \( \theta = \frac{60^\circ}{2} = 30^\circ \)

2. **Use the Formula for Resolving Power**:
The resolving power \( R_p \) of a microscope is given by the formula:
\[ R_p = \frac{2 \sin \theta}{1.22 \lambda} \]

3. **Calculate \( \sin \theta \)**:
For \( \theta = 30^\circ \):
\[ \sin 30^\circ = \frac{1}{2} \]

4. **Substitute Values into the Resolving Power Formula**:
Now substituting \( \sin 30^\circ \) and \( \lambda \) into the formula:
\[ R_p = \frac{2 \times \frac{1}{2}}{1.22 \times 5 \times 10^{-7}} \]

5. **Simplify the Expression**:
\[ R_p = \frac{1}{1.22 \times 5 \times 10^{-7}} = \frac{1}{6.1 \times 10^{-7}} \approx 1.639 \times 10^{6} \, \text{m}^{-1} \]

6. **Final Result**:
Therefore, the resolving power of the microscope is:
\[ R_p \approx 1.639 \times 10^{6} \, \text{m}^{-1} \]