A biconvex lens has a radius of curvature of magnitude `20cm`. Which one of the following options describes best the image formed of an object of height `2cm` place `30 cm` from the lens ?

Options

A. Virtual , upright, height `=0.5cm`
B. Real, inverted, height `=4 cm`
C. Real, inverted, height `=1 cm`
D. Virtual, upright, height `=1cm`

Explanation

To solve the problem of determining the nature and size of the image formed by a biconvex lens, we will follow these steps:

### Step 1: Identify the given data
- Radius of curvature (R) = 20 cm
- Object height (h_o) = 2 cm
- Object distance (u) = -30 cm (negative because the object is placed on the same side as the incoming light)

### Step 2: Calculate the focal length (f) of the lens
For a biconvex lens, the focal length (f) can be calculated using the radius of curvature. The formula for the focal length of a lens is given by:
\[ f = \frac{R}{2} \]
Since the lens is biconvex, the focal length is positive.
\[ f = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 3: Use the lens formula to find the image distance (v)
The lens 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:
\[ \frac{1}{v} = \frac{1}{10} + \frac{1}{-30} \]
Finding a common denominator (30):
\[ \frac{1}{v} = \frac{3}{30} - \frac{1}{30} = \frac{2}{30} = \frac{1}{15} \]
Thus,
\[ v = 15 \, \text{cm} \]

### Step 4: Calculate the magnification (M)
The magnification (M) is given by the formula:
\[ M = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the values:
\[ M = -\frac{15}{-30} = \frac{1}{2} \]
This means the image height (h_i) can be calculated as:
\[ h_i = M \cdot h_o = \frac{1}{2} \cdot 2 \, \text{cm} = 1 \, \text{cm} \]

### Step 5: Determine the nature of the image
Since the magnification is positive, this indicates that the image is virtual and upright.

### Conclusion
The image formed is virtual, upright, and has a height of 1 cm.

### Summary of the Answer
- **Nature of Image**: Virtual and upright
- **Height of Image**: 1 cm