An object is placed at a distance of 40 cm from a convex mirror fo focal length 40cm. Find image position and its magnification.Also ,draw its ray diagram.

Explanation

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

### Step 1: Identify the given values
- Object distance (u) = -40 cm (negative as per the sign convention for mirrors)
- Focal length (f) = +40 cm (positive for a convex mirror)

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

Substituting the known values into the formula:
\[ \frac{1}{40} = \frac{1}{-40} + \frac{1}{v} \]

### Step 3: Solve for image distance (v)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{40} + \frac{1}{40} \]
\[ \frac{1}{v} = \frac{2}{40} \]
\[ \frac{1}{v} = \frac{1}{20} \]
Thus,
\[ v = 20 \text{ cm} \]

### Step 4: Determine the nature of the image
Since the image distance \( v \) is positive, this indicates that the image is formed behind the mirror (virtual image).

### Step 5: Calculate the magnification (m)
The magnification (m) for a mirror is given by the formula:
\[ m = -\frac{v}{u} \]
Substituting the values we have:
\[ m = -\frac{20}{-40} = \frac{1}{2} = 0.5 \]
This means the image is half the size of the object.

### Step 6: Draw the ray diagram
1. Draw the principal axis and the convex mirror.
2. Mark the object at a distance of 40 cm from the mirror.
3. Draw a ray parallel to the principal axis from the top of the object. After reflecting off the mirror, it will appear to diverge from the focal point behind the mirror.
4. Draw a second ray from the top of the object towards the focal point. After reflecting, it will travel parallel to the principal axis.
5. The point where the reflected rays appear to diverge from is the location of the virtual image, which is 20 cm behind the mirror.

### Summary of Results
- Image position: 20 cm behind the mirror
- Magnification: 0.5 (the image is half the size of the object)