A microscope is used to resolved two objects placed at a distance of `2 xx 10^(-4) cm`. Calculate the numerical aperture of the microscope if light of wavelength `6,000 Å` is used.

Answer

To solve the problem of calculating the numerical aperture of a microscope given the distance between two objects and the wavelength of light used, we can follow these steps:

### Step 1: Understand the given data
We are given:
- The distance between two objects, \( d = 2 \times 10^{-4} \) cm.
- The wavelength of light used, \( \lambda = 6000 \) Å.

### Step 2: Convert the units
Convert the distance \( d \) from centimeters to meters:
\[ d = 2 \times 10^{-4} \, \text{cm} = 2 \times 10^{-4} \times 10^{-2} \, \text{m} = 2 \times 10^{-6} \, \text{m} \]

Convert the wavelength \( \lambda \) from angstroms to meters:
\[ \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]

### Step 3: Use the formula for limit of resolution
The limit of resolution \( d \) for a microscope is given by the formula:
\[ d = \frac{1.22 \lambda}{2 \mu \sin \theta} \]
where \( \mu \sin \theta \) is the numerical aperture (NA) of the microscope.

### Step 4: Rearrange the formula to find numerical aperture
Rearranging the formula to find the numerical aperture:
\[ \mu \sin \theta = \frac{1.22 \lambda}{2d} \]

### Step 5: Substitute the values into the formula
Substituting the values of \( \lambda \) and \( d \):
\[ \mu \sin \theta = \frac{1.22 \times (6 \times 10^{-7})}{2 \times (2 \times 10^{-6})} \]

### Step 6: Calculate the numerical aperture
Calculating the right-hand side:
\[ \mu \sin \theta = \frac{1.22 \times 6 \times 10^{-7}}{4 \times 10^{-6}} = \frac{7.32 \times 10^{-7}}{4 \times 10^{-6}} = 0.183 \]

### Final Answer
Thus, the numerical aperture of the microscope is:
\[ \text{Numerical Aperture} = 0.183 \]

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