The diameter of the sun is `1.4 X 10^9m` and its distance from the earth is `1.5 X 10^11 m`. Find the radius of the image of the sun formed by a lens of focal length 20 cm.

Explanation

To find the radius of the image of the sun formed by a lens, we will follow these steps:

### Step 1: Identify the given values
- Diameter of the sun (D) = \(1.4 \times 10^9 \, \text{m}\)
- Distance from the earth to the sun (u) = \(1.5 \times 10^{11} \, \text{m}\)
- Focal length of the lens (f) = \(20 \, \text{cm} = 20 \times 10^{-2} \, \text{m}\)

### Step 2: Convert the focal length to meters
- Focal length \(f = 20 \times 10^{-2} \, \text{m} = 0.2 \, \text{m}\)

### Step 3: Determine the object distance
Since the distance of the sun from the earth is very large compared to the focal length of the lens, we can consider the object distance (u) to be at infinity. Therefore, the image will be formed at the focal point of the lens.

### Step 4: Use the lens formula
For an object at infinity, the image distance (v) is equal to the focal length (f):
- \(v = f = 0.2 \, \text{m}\)

### Step 5: Calculate the magnification
The magnification (m) is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Where:
- \(h'\) = height of the image
- \(h\) = height of the object (which is the diameter of the sun)

### Step 6: Calculate the diameter of the image
Using the magnification formula:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{0.2}{1.5 \times 10^{11}} \]

Now, the diameter of the image (D') can be found using:
\[ D' = m \cdot D \]
Where \(D\) is the diameter of the sun.

Substituting the values:
\[ D' = -\frac{0.2}{1.5 \times 10^{11}} \cdot (1.4 \times 10^9) \]

### Step 7: Calculate the diameter of the image
Calculating the above expression:
\[ D' = -\frac{0.2 \times 1.4 \times 10^9}{1.5 \times 10^{11}} \]
\[ D' = -\frac{0.28 \times 10^9}{1.5 \times 10^{11}} \]
\[ D' = -1.86 \times 10^{-2} \, \text{m} \]

### Step 8: Calculate the radius of the image
The radius of the image (r') is half of the diameter:
\[ r' = \frac{D'}{2} = \frac{-1.86 \times 10^{-2}}{2} \]
\[ r' = -0.93 \times 10^{-2} \, \text{m} = -0.00093 \, \text{m} = -0.93 \, \text{mm} \]

### Final Answer
The radius of the image of the sun formed by the lens is approximately \(0.93 \, \text{mm}\).

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