A source of light and a screen are placed 90 cm apart. Where should a convex lens of 20 cm focal length be placed in order to form a real image of the source on the screen ?

Answer

To solve the problem of where to place a convex lens of 20 cm focal length to form a real image on a screen that is 90 cm away from the light source, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the distance from the light source to the lens be \( u \) (object distance).
- Let the distance from the lens to the screen be \( v \) (image distance).
- The total distance between the light source and the screen is given as 90 cm, so we can express this as:
\[ u + v = 90 \text{ cm} \]

2. **Use the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length of the lens. For a convex lens, the focal length \( f = 20 \) cm.

3. **Express \( v \) in terms of \( u \)**:
From the equation \( u + v = 90 \), we can express \( v \) as:
\[ v = 90 - u \]

4. **Substitute \( v \) in the Lens Formula**:
Substitute \( v \) into the lens formula:
\[ \frac{1}{20} = \frac{1}{90 - u} - \frac{1}{u} \]

5. **Clear the Fractions**:
To eliminate the fractions, we can multiply through by \( 20u(90 - u) \):
\[ u(90 - u) = 20(90 - u) - 20u \]

6. **Simplify the Equation**:
Expanding and simplifying gives:
\[ 90u - u^2 = 1800 - 20u - 20u \]
\[ 90u - u^2 = 1800 - 40u \]
Rearranging this leads to:
\[ u^2 - 130u + 1800 = 0 \]

7. **Solve the Quadratic Equation**:
We can use the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -130 \), and \( c = 1800 \).
\[ u = \frac{130 \pm \sqrt{(-130)^2 - 4 \cdot 1 \cdot 1800}}{2 \cdot 1} \]
\[ u = \frac{130 \pm \sqrt{16900 - 7200}}{2} \]
\[ u = \frac{130 \pm \sqrt{10100}}{2} \]
\[ u = \frac{130 \pm 100.5}{2} \]

This gives us two possible values for \( u \):
\[ u_1 = \frac{230.5}{2} = 115.25 \text{ cm} \quad \text{(not valid since } u + v = 90 \text{)} \]
\[ u_2 = \frac{29.5}{2} = 14.75 \text{ cm} \quad \text{(valid)} \]

8. **Calculate \( v \)**:
Using \( u + v = 90 \):
\[ v = 90 - 14.75 = 75.25 \text{ cm} \]

9. **Conclusion**:
The lens should be placed at a distance of approximately \( 14.75 \) cm from the light source to form a real image on the screen.