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: Problem Definition:
- Object height: 4 cm.
- Convex lens focal length: \( f = 24 \, \text{cm} \).
- Object distance: \( u = -16 \, \text{cm} \) (real object).
Objective: Determine the image position and size using the lens formula.

Step 2: Lens Formula:
The governing equation is:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Where:
- \( f \): focal length.
- \( v \): image distance.
- \( u \): object distance.

Step 3: Applying Given Values:
Input parameters:
- \( f = 24 \, \text{cm} \).
- \( u = -16 \, \text{cm} \).

Substitution into the lens formula:
\[ \frac{1}{24} = \frac{1}{v} - \frac{1}{-16} \] Solving for \( v \):
\[ \frac{1}{v} = \frac{1}{24} + \frac{1}{16} \] Using the least common multiple (48) for 24 and 16:
\[ \frac{1}{v} = \frac{2}{48} + \frac{3}{48} = \frac{5}{48} \] Therefore, \( v = \frac{48}{5} = 9.6 \, \text{cm} \).

The image is located 9.6 cm from the lens. A positive \( v \) indicates a real image formed on the opposite side of the object.

Step 4: Image Size Calculation:
Magnification formula:
\[ \text{Magnification} (M) = \frac{\text{Image height}}{\text{Object height}} = \frac{v}{u} \] Substituting known values:
\[ M = \frac{9.6}{-16} = -0.6 \] The negative magnification signifies an inverted image. Image height is calculated as:
\[ \text{Image height} = \text{Object height} \times M = 4 \times (-0.6) = -2.4 \, \text{cm} \] The negative sign confirms image inversion.

Step 5: Summary of Results:
- Image distance from lens: 9.6 cm.
- Image size: 2.4 cm (inverted).