An object is placed at a distance of 30 cm from a concave mirror of
focal length 20 cm .find image distance and its magnification. Also ,draw the ray
diagram.

Options

A. +2
B. +4
C. -`1/2`
D. -2

Explanation

To solve the problem step by step, let's follow the method outlined in the video transcript:

### Step 1: Identify the Given Values
- Object distance (U) = -30 cm (negative because the object is in front of the mirror)
- Focal length (F) = -20 cm (negative for concave mirrors)

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

### Step 3: Substitute the Values into the Mirror Formula
Substituting the known values into the formula:
\[ \frac{1}{V} + \frac{1}{-30} = \frac{1}{-20} \]

### Step 4: Rearrange the Equation
Rearranging gives:
\[ \frac{1}{V} = \frac{1}{-20} + \frac{1}{30} \]

### Step 5: Find a Common Denominator and Simplify
The common denominator for -20 and 30 is 60. Thus:
\[ \frac{1}{-20} = \frac{-3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \]
So,
\[ \frac{1}{V} = \frac{-3 + 2}{60} = \frac{-1}{60} \]

### Step 6: Solve for V
Taking the reciprocal gives:
\[ V = -60 \, \text{cm} \]
This means the image is formed 60 cm in front of the mirror (on the same side as the object).

### Step 7: Calculate Magnification
The magnification (m) is given by the formula:
\[ m = -\frac{V}{U} \]
Substituting the values:
\[ m = -\frac{-60}{-30} = -2 \]

### Step 8: Interpret the Magnification
The negative sign indicates that the image is inverted, and the magnitude of 2 means the image is twice the size of the object.

### Step 9: Draw the Ray Diagram
1. Draw the concave mirror with the focal point (F) at 20 cm and the center of curvature (C) at 40 cm.
2. Place the object (O) at 30 cm in front of the mirror.
3. Draw three rays:
- A ray parallel to the principal axis that reflects through the focal point.
- A ray passing through the focal point that reflects parallel to the principal axis.
- A ray striking the mirror at the vertex that reflects back at the same angle.
4. The intersection of the reflected rays indicates the position of the image (I) at 60 cm in front of the mirror.

### Final Results
- Image distance (V) = -60 cm
- Magnification (m) = -2 (inverted and double the size)