A concave mirror of radius of curvature 20 cm forms image of the sun. The diameter of the sun subtends an angle `1^(@)` on the earth. Then the diameter of the image is (in cm) :

Options

A. `2 pi//9`
B. `pi//9`
C. 20
D. `pi//18`

Answer

To solve the problem step by step, we will follow the concepts of geometrical optics, particularly focusing on concave mirrors and the relationships between object distance, image distance, and focal length.

### Step 1: Understand the Given Information
- **Radius of curvature (R)** of the concave mirror = 20 cm
- **Angle subtended by the sun (θ)** = 1 degree
- **Object distance (u)** for the sun is considered to be very large (infinity).

### Step 2: Calculate the Focal Length (f)
The focal length (f) of a concave mirror is given by the formula:
\[ f = -\frac{R}{2} \]
Substituting the given radius of curvature:
\[ f = -\frac{20 \, \text{cm}}{2} = -10 \, \text{cm} \]

### Step 3: Apply the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( v \) = image distance
- \( u \) = object distance

Since the sun is very far away, we can consider \( u \) to be infinity:
\[ \frac{1}{f} = \frac{1}{v} + 0 \]
Thus, we have:
\[ \frac{1}{v} = \frac{1}{f} \]
Substituting the focal length:
\[ \frac{1}{v} = \frac{1}{-10} \]
This gives:
\[ v = -10 \, \text{cm} \]

### Step 4: Convert the Angle from Degrees to Radians
To calculate the diameter of the image, we need to convert the angle from degrees to radians:
\[ \text{Angle in radians} = \theta = \frac{\pi}{180} \, \text{radians} \]

### Step 5: Relate the Diameter of the Sun to the Focal Length
Using the formula that relates the angle subtended by an object to its diameter and focal length:
\[ \theta = \frac{D}{f} \]
Where:
- \( D \) = diameter of the sun
- \( f \) = focal length

Rearranging this gives us:
\[ D = \theta \cdot f \]
Substituting the values:
\[ D = \left(\frac{\pi}{180}\right) \cdot 10 \, \text{cm} \]
\[ D = \frac{10\pi}{180} \]
\[ D = \frac{\pi}{18} \, \text{cm} \]

### Step 6: Calculate the Diameter of the Image
Since the image is formed at the focal point, the diameter of the image will be the same as the diameter of the sun calculated above:
\[ \text{Diameter of the image} = \frac{\pi}{18} \, \text{cm} \]

### Final Answer
The diameter of the image is approximately \( \frac{\pi}{18} \) cm or about 0.174 cm.

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