A lens of focal length 20.0 cm and aperture radius 2.0cm is placed at a distance 30.cm from a point source of light. On the other side a screen is placed at a distance 50.0cm from the lens. The radius of spot of light formed on screen is (neglect spherical abberation through lens)

Options

A. `1//2cm`
B. `1//3cm`
C. `1//5cm`
D. `1.0cm`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the lens formula
The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. The formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

### Step 2: Identify the given values
From the problem, we have:
- Focal length \( f = 20 \, \text{cm} \)
- Object distance \( u = -30 \, \text{cm} \) (negative because the object is on the same side as the incoming light)
- Aperture radius \( R = 2 \, \text{cm} \)

### Step 3: Substitute the values into the lens formula
We need to find the image distance \( v \). Rearranging the lens formula gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{-30} \]

### Step 4: Calculate the right-hand side
To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{-30} = \frac{-2}{60} \]
So,
\[ \frac{1}{v} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

### Step 5: Find the image distance \( v \)
Taking the reciprocal gives:
\[ v = 60 \, \text{cm} \]

### Step 6: Determine the position of the image
Since the screen is placed at a distance of 50 cm from the lens, and the image distance \( v = 60 \, \text{cm} \) indicates that the image is formed 10 cm behind the screen.

### Step 7: Calculate the radius of the spot of light on the screen
Using similar triangles, we can find the radius of the spot of light on the screen. The triangles formed by the rays of light passing through the lens and hitting the screen are similar.

Let \( h \) be the radius of the spot of light on the screen. From the similar triangles:
\[ \frac{h}{10} = \frac{2}{60} \]
Cross-multiplying gives:
\[ h = \frac{2}{60} \times 10 \]
\[ h = \frac{20}{60} = \frac{1}{3} \, \text{cm} \]

### Final Answer
The radius of the spot of light formed on the screen is:
\[ \frac{1}{3} \, \text{cm} \]