A particle goes in a circle of radius 2.0 cm. A concave mirror of focal length 20 cm is placed with its principal axis passing through the centre of the circle and perpendicular to its plane. The distance between the pole of the mirror and the centre of the circle is 30 cm. Calculate the radius of the circle formed by the image.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Setup
We have a concave mirror with a focal length of \( f = -20 \, \text{cm} \) (the negative sign indicates that it is a concave mirror). The distance from the mirror to the center of the circle (the object) is \( u = -30 \, \text{cm} \) (the negative sign indicates that the object is in front of the mirror).

### Step 2: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance.

Substituting the known values into the formula:
\[ \frac{1}{-20} = \frac{1}{v} + \frac{1}{-30} \]

### Step 3: Solve for \( v \)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-20} + \frac{1}{30} \]
Finding a common denominator (which is 60):
\[ \frac{1}{v} = \frac{-3}{60} + \frac{2}{60} = \frac{-1}{60} \]
Thus,
\[ v = -60 \, \text{cm} \]

### Step 4: Calculate Magnification
The magnification \( m \) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{-60}{-30} = 2 \]

### Step 5: Calculate the Height of the Image
The height of the object is given as \( h_0 = 2 \, \text{cm} \). The height of the image \( h_i \) can be calculated using:
\[ h_i = m \cdot h_0 \]
Substituting the values:
\[ h_i = 2 \cdot 2 = 4 \, \text{cm} \]

### Step 6: Determine the Radius of the Circle Formed by the Image
Since the image is formed at a distance of \( v = -60 \, \text{cm} \) and the height of the image is \( 4 \, \text{cm} \), the radius of the circle formed by the image is equal to the height of the image, which is:
\[ \text{Radius of the circle formed by the image} = 4 \, \text{cm} \]

### Final Answer
The radius of the circle formed by the image is \( 4 \, \text{cm} \).

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