An object of `1cm^(2)` face area is placed at a distance of 1.5m from a screen. How far from the object should a concave mirror be placed so that it forms `4 cm^(2)` image of object on the screen? Also, calculate the focal length of the mirror.

Answer

To solve the problem step by step, we will follow the concepts of optics, specifically related to concave mirrors and magnification.

### Step 1: Understand the given data
- Area of the object = \(1 \, \text{cm}^2\)
- Area of the image = \(4 \, \text{cm}^2\)
- Distance from the object to the screen = \(1.5 \, \text{m} = 150 \, \text{cm}\)

### Step 2: Calculate the magnification
The magnification \(m\) is given by the ratio of the area of the image to the area of the object:
\[ m = \frac{\text{Area of image}}{\text{Area of object}} = \frac{4 \, \text{cm}^2}{1 \, \text{cm}^2} = 4 \]

### Step 3: Relate magnification to distances
The magnification for mirrors is also given by the formula:
\[ m = -\frac{v}{u} \]
Where:
- \(v\) = image distance from the mirror
- \(u\) = object distance from the mirror

From the magnification we calculated:
\[ 4 = -\frac{v}{u} \implies v = -4u \]

### Step 4: Set up the distance equation
The total distance from the object to the screen is the sum of the object distance and the image distance:
\[ u + v = 150 \, \text{cm} \]
Substituting \(v\) from the magnification equation:
\[ u - 4u = 150 \implies -3u = 150 \implies u = -50 \, \text{cm} \]
Since \(u\) is negative (as per the sign convention for mirrors), we take the absolute value:
\[ u = 50 \, \text{cm} \]

### Step 5: Calculate the image distance
Using \(v = -4u\):
\[ v = -4(-50) = 200 \, \text{cm} \]

### Step 6: Find the distance of the mirror from the object
The distance of the mirror from the object is:
\[ \text{Distance from object to mirror} = |u| = 50 \, \text{cm} \]

### Step 7: Calculate the focal length of the mirror
Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{f} = \frac{1}{200} + \frac{1}{-50} \]
Finding a common denominator (200):
\[ \frac{1}{f} = \frac{1}{200} - \frac{4}{200} = \frac{-3}{200} \]
Thus:
\[ f = -\frac{200}{3} \approx -66.67 \, \text{cm} \]

### Summary of Results
- Distance from the object to the mirror: \(50 \, \text{cm}\)
- Focal length of the mirror: \(-66.67 \, \text{cm}\)