To solve the problem of finding the resolving power of a microscope given the parameters, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Given Values:**
- Focal length, \( f = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Wavelength, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \)
- Aperture (diameter), \( a = 1 \, \text{cm} = 0.01 \, \text{m} \)
2. **Use the Formula for Resolving Power (RP):**
The resolving power of a microscope is given by the formula:
\[
RP = \frac{2 \mu \sin \theta}{1.22 \lambda}
\]
where \( \mu \) is the refractive index of the medium (for air, \( \mu = 1 \)).
3. **Calculate \( \sin \theta \):**
For small angles, we can use the approximation:
\[
\tan \theta \approx \sin \theta \approx \frac{a}{f}
\]
Substituting the values:
\[
\sin \theta = \frac{a}{f} = \frac{0.01 \, \text{m}}{0.05 \, \text{m}} = \frac{1}{5}
\]
4. **Substitute Values into the Resolving Power Formula:**
Now substitute \( \mu = 1 \), \( \sin \theta = \frac{1}{5} \), and \( \lambda = 6 \times 10^{-7} \, \text{m} \) into the resolving power formula:
\[
RP = \frac{2 \times 1 \times \frac{1}{5}}{1.22 \times 6 \times 10^{-7}}
\]
5. **Calculate the Resolving Power:**
First, calculate the numerator:
\[
2 \times 1 \times \frac{1}{5} = \frac{2}{5} = 0.4
\]
Now calculate the denominator:
\[
1.22 \times 6 \times 10^{-7} = 7.32 \times 10^{-7}
\]
Now, substituting back into the equation:
\[
RP = \frac{0.4}{7.32 \times 10^{-7}} \approx 5.46 \times 10^{5} \, \text{per meter}
\]
6. **Final Result:**
The resolving power of the microscope is approximately:
\[
RP \approx 5.46 \times 10^{5} \, \text{per meter}
\]
