Question

In an experiment with a convex lens, the length of an image is 1 cm, and the object length is 5 cm. If the object is placed at a distance of 40 cm from the lens, then the focal length of the lens is

• A. 6.67 cm
• B. 13.5 cm
• C. 5.6 cm
• D. 3.6 cm

Hint:

Applying the sign convention correctly is crucial. For real images formed by a single convex lens, the image is inverted, so \(h_i\) is negative. This leads to a negative magnification \(m\), and the image distance \(v\) will be positive.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires using the lens formula and the magnification formula for a convex lens. We can find the image distance using the magnification and then use the lens formula to calculate the focal length.

Step 2: Key Formula or Approach:
1. Magnification, \(m = \frac{\text{height of image } (h_i)}{\text{height of object } (h_o)} = \frac{\text{image distance } (v)}{\text{object distance } (u)}\).
2. Lens Formula: \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\).
3. New Cartesian Sign Convention: Light travels from left to right. Distances measured against the incident light are negative. Distances in the direction of light are positive. Object distance \(u\) is negative.

Step 3: Detailed Explanation:
Given data:
Height of object, \(h_o = 5 \, \text{cm}\).
Height of image, \(h_i = 1 \, \text{cm}\).
Object distance, \(u = -40 \, \text{cm}\) (by sign convention).
Since the image is smaller than the object (\(h_i < h_o\)), the image formed by the convex lens must be real and inverted. Therefore, the image height should be taken as negative. \(h_i = -1 \, \text{cm}\).
Part 1: Calculate the image distance (v).
Using the magnification formula: \[ m = \frac{h_i}{h_o} = \frac{-1 \, \text{cm}}{5 \, \text{cm}} = -\frac{1}{5} \] Also, \(m = \frac{v}{u}\). \[ -\frac{1}{5} = \frac{v}{-40 \, \text{cm}} \] \[ v = (-40 \, \text{cm}) \times \left(-\frac{1}{5}\right) = +8 \, \text{cm} \] The positive sign for \(v\) confirms that a real image is formed on the opposite side of the lens.
Part 2: Calculate the focal length (f).
Using the lens formula: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] \[ \frac{1}{f} = \frac{1}{8} - \frac{1}{-40} = \frac{1}{8} + \frac{1}{40} \] To add the fractions, find a common denominator, which is 40. \[ \frac{1}{f} = \frac{5}{40} + \frac{1}{40} = \frac{6}{40} = \frac{3}{20} \] \[ f = \frac{20}{3} \, \text{cm} \approx 6.67 \, \text{cm} \]

Step 4: Final Answer:
The focal length of the lens is 6.67 cm.