An object 0.04 m high is placed at a distance of 0.8 m from a concave mirror of radius of curvature 0.4 m. find the position, nature and the size of the image formed.

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the Focal Length of the Concave Mirror
The radius of curvature (R) of the concave mirror is given as 0.4 m. The focal length (F) can be calculated using the formula:
\[ F = \frac{R}{2} \]
Substituting the value:
\[ F = \frac{0.4 \, \text{m}}{2} = 0.2 \, \text{m} \]

### Step 2: Convert Units
Convert the focal length and object distance into centimeters for easier calculations:
- Focal length, \( F = 0.2 \, \text{m} = 20 \, \text{cm} \)
- Object distance, \( U = -0.8 \, \text{m} = -80 \, \text{cm} \) (the negative sign indicates that the object is in front of the mirror)

### Step 3: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-80} \]

### Step 4: Solve for Image Distance (V)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{80} \]
Finding a common denominator (80):
\[ \frac{1}{v} = \frac{4}{80} - \frac{1}{80} = \frac{3}{80} \]
Thus,
\[ v = \frac{80}{3} \approx 26.67 \, \text{cm} \]
Since this value is positive, it indicates that the image is formed on the same side as the object.

### Step 5: Determine the Nature of the Image
Since the image distance (V) is positive, the image is real and inverted.

### Step 6: Calculate the Magnification
The magnification (M) is given by the formula:
\[ M = -\frac{v}{u} \]
Substituting the known values:
\[ M = -\frac{26.67}{-80} = \frac{26.67}{80} \approx 0.333 \]

### Step 7: Calculate the Size of the Image
The size of the image (h') can be calculated using:
\[ h' = M \cdot h \]
Where \( h \) is the height of the object (0.04 m):
\[ h' = 0.333 \times 0.04 \approx 0.0133 \, \text{m} = 1.33 \, \text{cm} \]

### Final Results
- **Position of the Image**: 26.67 cm (real and inverted)
- **Nature of the Image**: Real and inverted
- **Size of the Image**: 1.33 cm