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 follow these steps:

### Step 1: Identify the given values
- Focal length of the convex mirror (F) = +40 cm (positive because it is a convex mirror)
- Object distance (U) = -40 cm (negative as per sign convention for mirrors)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
We need to find the image distance (V).

### Step 3: Rearranging the formula
Rearranging the mirror formula to find V:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]

### Step 4: Substitute the known values
Substituting the values of F and U into the equation:
\[ \frac{1}{v} = \frac{1}{40} - \frac{1}{-40} \]
This simplifies to:
\[ \frac{1}{v} = \frac{1}{40} + \frac{1}{40} = \frac{2}{40} = \frac{1}{20} \]

### Step 5: Calculate V
Taking the reciprocal to find V:
\[ v = 20 \text{ cm} \]
This means the image is formed at a distance of 20 cm on the positive side (right side) of the mirror.

### Step 6: Calculate the magnification (m)
The magnification (m) is given by the formula:
\[ m = -\frac{v}{u} \]
Substituting the values of V and U:
\[ m = -\frac{20}{-40} = \frac{20}{40} = 0.5 \]

### Step 7: Interpret the magnification
Since the magnification is positive (0.5), it indicates that the image is virtual and upright. The size of the image is half that of the object.

### Step 8: Draw the ray diagram
1. Draw the convex mirror and mark the focal point (F) at 40 cm on the right side.
2. Place the object (O) at 40 cm on the left side.
3. Draw two rays from the top of the object:
- One ray parallel to the principal axis that reflects through the focal point.
- Another ray directed towards the focal point that reflects parallel to the principal axis.
4. Extend the reflected rays backward to locate the image (I) at 20 cm on the right side of the mirror.

### Final Results
- Image position (V) = 20 cm (on the right side of the mirror)
- Magnification (m) = 0.5 (image is upright and half the size of the object)

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