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

Explanation

To solve the problem step by step, we will use the mirror formula and the magnification formula for a concave mirror.

### 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 a concave mirror)

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

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

### Step 3: Solve for image distance (v)
Rearranging the equation to isolate \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{-20} - \frac{1}{-30} \]

Finding a common denominator (which is 60):
\[ \frac{1}{v} = \frac{-3}{60} + \frac{2}{60} = \frac{-1}{60} \]

Now, taking the reciprocal to find \( v \):
\[ v = -60 \text{ cm} \]

### Step 4: Interpret the image distance
The negative sign indicates that the image is formed in front of the mirror, which means it is a real image.

### Step 5: Calculate magnification (m)
The magnification (m) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{-60}{-30} = -2 \]

### Step 6: 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.

### Summary of Results
- Image distance (v) = -60 cm (real and inverted)
- Magnification (m) = -2 (image is twice the size of the object)

### Step 7: Draw the ray diagram
1. Draw a concave mirror with its principal axis.
2. Mark the focal point (F) at 20 cm from the mirror.
3. Place the object (O) at 30 cm in front of the mirror.
4. Draw three rays:
- The first ray parallel to the principal axis, reflecting through the focal point.
- The second ray passing through the focal point and reflecting parallel to the principal axis.
- The third ray directed towards the mirror's pole, reflecting off at the same angle.
5. The intersection of the reflected rays gives the location of the image.