A telescope has an objective lens of `10cm` diameter and is situated at a distance of one kilometre from two objects. The minimum distance between these two objects, which can be resolved by the telescope, when the mean wavelength of light is `5000 Å`, of the order of

Options

A. `0.5` m
B. `5` m
C. `5` mm
D. `5` cm

Answer

To solve the problem, we need to determine the minimum distance between two objects that can be resolved by a telescope with a given objective lens diameter and wavelength of light. We will use the formula for the resolving power of a telescope.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Diameter of the objective lens (D) = 10 cm = 0.1 m
- Distance to the objects (d) = 1 km = 1000 m
- Wavelength of light (λ) = 5000 Å = 5000 × 10^(-10) m = 5 × 10^(-7) m

2. **Use the Resolving Power Formula:**
The resolving power (θ) of a telescope is given by the formula:
\[ \theta = \frac{\lambda}{D} \]
where:
- θ is the angular resolution in radians,
- λ is the wavelength of light,
- D is the diameter of the lens.

3. **Calculate the Angular Resolution (θ):**
Substitute the values into the formula:
\[ \theta = \frac{5 \times 10^{-7} \text{ m}}{0.1 \text{ m}} = 5 \times 10^{-6} \text{ radians} \]

4. **Calculate the Minimum Distance (x) Between the Two Objects:**
The minimum distance (x) that can be resolved is given by:
\[ x = \theta \cdot d \]
Substitute θ and d:
\[ x = (5 \times 10^{-6} \text{ radians}) \cdot (1000 \text{ m}) = 5 \times 10^{-3} \text{ m} = 5 \text{ mm} \]

5. **Final Result:**
The minimum distance between the two objects that can be resolved by the telescope is **5 mm**.