Question

Obtain a lens combination with the specified focal length using two lenses from the given lens.

Hint:

The formula \( \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \) is valid for lenses in contact. If there is a distance 'd' between the lenses, the formula becomes more complex: \( \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \). For this experiment, ensure the lenses are as close together as possible.

Answer 1

Step 1: Understanding the Concept:
When two thin lenses are placed in contact, they behave as a single lens with an equivalent focal length (F) that depends on the individual focal lengths (\(f_1\) and \(f_2\)) of the two lenses. By choosing the right combination of lenses, a combination with a desired focal length can be created.
Step 2: Key Formula and Apparatus:
Apparatus Required:
- A collection of lenses (at least two, e.g., two convex lenses or one convex and one concave). - An optical bench with lens holders. - An object (like an illuminated screen or an optical needle). - A screen to view the image.
Key Formula:
The formula for the equivalent focal length (F) of two thin lenses in contact is: \[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \] The power of the combination is \(P = P_1 + P_2\).
Step 3: Detailed Procedure:
1. Measure Individual Focal Lengths: - First, determine the focal length of each of the two given lenses (\(f_1\) and \(f_2\)) individually. This can be done by finding the approximate focal length (focusing a distant object) or more accurately using the u-v method on an optical bench. Remember that focal length is positive for a convex lens and negative for a concave lens.
2. Calculate Theoretical Combined Focal Length: - Let's say you have two convex lenses with \(f_1 = +20\) cm and \(f_2 = +30\) cm. - The theoretical focal length of the combination would be: \[ \frac{1}{F} = \frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{5}{60} = \frac{1}{12} \] \[ F = +12 \text{ cm} \] - If one lens is convex (\(f_1 = +20\) cm) and one is concave (\(f_2 = -30\) cm): \[ \frac{1}{F} = \frac{1}{20} - \frac{1}{30} = \frac{3 - 2}{60} = \frac{1}{60} \] \[ F = +60 \text{ cm} \] 3. Verify Experimental Combined Focal Length: - Place the two lenses in contact with each other in a single holder on the optical bench. Treat this combination as a single lens. - Place an object pin at a known distance 'u' from the combination. - Locate the position of the real image formed on the other side using a screen or another needle (by removing parallax). Measure the image distance 'v'. - Use the lens formula \( \frac{1}{F_{exp}} = \frac{1}{v} - \frac{1}{u} \) to calculate the experimental focal length of the combination.
Step 4: Result:
Compare the experimentally determined focal length (\(F_{exp}\)) with the theoretically calculated focal length (F). They should be approximately equal. This demonstrates how to obtain a lens combination with a specific, predictable focal length.