Question

A 4 cm tall object is placed perpendicular to the principal axis of a convex lens of focal length 24 cm. The distance of the object from the lens is 16 cm. Find the position and size of the image formed.

Answer 1

Step 1: Understanding the problem:
- The object is 4 cm tall.
- The convex lens has a focal length \( f = 24 \, \text{cm} \).
- The object distance \( u = -16 \, \text{cm} \) (object distance is always negative for real objects).
We need to find the position and size of the image formed using the lens formula.

Step 2: 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 from the lens.
- \( u \) is the object distance from the lens.

Step 3: Substituting the given values into the lens formula:
We are given:
- Focal length \( f = 24 \, \text{cm} \).
- Object distance \( u = -16 \, \text{cm} \).

Substitute these values into the lens formula:
\[ \frac{1}{24} = \frac{1}{v} - \frac{1}{-16} \] Now solve for \( v \):
\[ \frac{1}{v} = \frac{1}{24} + \frac{1}{16} \] Taking the LCM of 24 and 16, which is 48, we get:
\[ \frac{1}{v} = \frac{2}{48} + \frac{3}{48} = \frac{5}{48} \] Thus, \( v = \frac{48}{5} = 9.6 \, \text{cm} \).

So, the image is formed at a distance of 9.6 cm from the lens.
Since \( v \) is positive, the image is real and formed on the opposite side of the object.

Step 4: Calculating the size of the image:
To calculate the size of the image, we use the magnification formula:
\[ \text{Magnification} (M) = \frac{\text{Image height}}{\text{Object height}} = \frac{v}{u} \] Substitute the known values:
\[ M = \frac{9.6}{-16} = -0.6 \] The magnification is negative, indicating that the image is inverted. Now, calculate the image height:
\[ \text{Image height} = \text{Object height} \times M = 4 \times (-0.6) = -2.4 \, \text{cm} \] The negative sign indicates that the image is inverted.

Step 5: Conclusion:
- The image is formed at a distance of 9.6 cm from the lens.
- The size of the image is 2.4 cm, and it is inverted.