An object of height 5 cm is placed in front of a convex lens of focal length 20 cm at a distance of 30 cm. What is the height of the image (in cm)?

Options

A. 20
B. 30
C. 10
D. 40

Explanation

To find the height of the image formed by a convex lens, we can follow these steps:

### Step 1: Identify the given values
- Height of the object (h_o) = 5 cm
- Focal length of the lens (f) = +20 cm (positive for convex lens)
- Object distance (u) = -30 cm (object distance is taken as negative in lens formula)

### Step 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,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 3: Substitute the values into the lens formula
Substituting the known values into the lens formula:
\[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} \]
This simplifies to:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{30} \]

### Step 4: Rearranging the equation
To find \( \frac{1}{v} \), we can rearrange the equation:
\[ \frac{1}{v} = \frac{1}{20} - \frac{1}{30} \]

### Step 5: Find a common denominator and solve
The common denominator for 20 and 30 is 60. Thus,
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Now substituting these values:
\[ \frac{1}{v} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]
So,
\[ v = 60 \text{ cm} \]

### Step 6: Calculate magnification
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
Where:
- \( h_i \) is the height of the image,
- \( h_o \) is the height of the object.

### Step 7: Substitute the values into the magnification formula
Substituting the known values:
\[ m = \frac{60}{-30} = -2 \]

### Step 8: Calculate the height of the image
Now, we can find the height of the image:
\[ h_i = m \cdot h_o = -2 \cdot 5 = -10 \text{ cm} \]
The negative sign indicates that the image is inverted.

### Step 9: Conclusion
The height of the image is 10 cm (inverted).

### Final Answer
The height of the image is **10 cm**.

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