An object is placed in front of a concave mirror of radius of curvature `40 cm` at a distance of `10 cm`. Find the position, nature and magnification of the image.

Explanation

To solve the problem step by step, we will follow the principles of optics related to concave mirrors.

### Step 1: Determine the Focal Length
The radius of curvature (R) of the concave mirror is given as 40 cm. The focal length (f) of a concave mirror is given by the formula:
\[ f = -\frac{R}{2} \]
Substituting the value of R:
\[ f = -\frac{40 \, \text{cm}}{2} = -20 \, \text{cm} \]

### Step 2: Define the Object Distance
The object is placed at a distance (u) of 10 cm in front of the mirror. In optics, the object distance is taken as negative for concave mirrors:
\[ u = -10 \, \text{cm} \]

### Step 3: Apply the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging this gives:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
Substituting the values of f and u:
\[ \frac{1}{v} = \frac{1}{-20} - \frac{1}{-10} \]
This simplifies to:
\[ \frac{1}{v} = -\frac{1}{20} + \frac{1}{10} \]
Finding a common denominator (20):
\[ \frac{1}{v} = -\frac{1}{20} + \frac{2}{20} = \frac{1}{20} \]
Thus:
\[ v = 20 \, \text{cm} \]

### Step 4: Determine the Nature of the Image
Since the value of v is positive, this indicates that the image is formed on the same side as the object, which means the image is virtual.

### Step 5: Calculate the Magnification
The magnification (m) of a mirror is given by the formula:
\[ m = -\frac{v}{u} \]
Substituting the values of v and u:
\[ m = -\frac{20}{-10} = 2 \]

### Summary of Results
- **Position of the Image**: 20 cm from the mirror
- **Nature of the Image**: Virtual
- **Magnification**: 2 (indicating the image is erect and twice the size of the object)