A 5 cm tall object is placed perpendicular to the principle axis of a convex lens of focal length 20 cm. The distance of the object from lens is 30 cm. Then, position and size of image formed is

Options

A. V= - 60 cm and h = 10 cm
B. V= + 60 cm and h = 10 cm
C. V= - 30 cm and h = 5 cm
D. V= + 60 cm and h = 5 cm

Answer

To solve the problem step by step, we will use the lens formula and the magnification formula.

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

### Step 2: Use the lens formula
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
Rearranging it gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

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

### Step 4: Find a common denominator and calculate
The common denominator for 20 and -30 is 60. Thus:
\[ \frac{1}{v} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

### Step 5: Solve for v
Taking the reciprocal gives:
\[ v = 60 \text{ cm} \]
This means the image is formed 60 cm on the opposite side of the lens.

### Step 6: Calculate the magnification
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
Substituting the values:
\[ m = \frac{60}{-30} = -2 \]
The negative sign indicates that the image is inverted.

### Step 7: Find the height of the image
Using the magnification to find the height of the image (h_i):
\[ h_i = m \cdot h_o = -2 \cdot 5 \text{ cm} = -10 \text{ cm} \]
The negative sign indicates that the image is inverted.

### Conclusion
- Position of the image: 60 cm on the opposite side of the lens
- Size of the image: 10 cm (inverted)

### Summary
The image formed is located 60 cm from the lens on the opposite side, and its height is 10 cm (inverted).

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