Problem 103 - Fundamentals of Physics

Question

Two thin lenses of focal lengths \(f_{1}\) and \(f_{2}\) are in contact and share the same central axis. Show that, in image formation, they are equivalent to a single thin lens for which the focal length is \(f=f_{1} f_{2} /\left(f_{1}+f_{2}\right)\)

Step-by-Step Solution

Answer

The focal length of the lens system is \( f = \frac{f_1 f_2}{f_1 + f_2} \).

Step 1: Understanding the Lens Formula

Recall the lens formula for a single lens: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. For two lenses in contact, the system can be thought of as having a combined effect on light rays.

Step 2: Combining the Lens Equations

When two lenses are in contact, their powers (the reciprocals of their focal lengths) are additive. Therefore, the combined power \( P \) of the system is given by \( P = P_1 + P_2 \), where \( P_1 = \frac{1}{f_1} \) and \( P_2 = \frac{1}{f_2} \).

Step 3: Calculating the Combined Focal Length

Substitute the powers \( P_1 \) and \( P_2 \) into the equation for combined power: \( P = \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \).

Step 4: Solving for the Combined Focal Length

Now, find \( f \) by taking the reciprocal of the combined power: \( f = \frac{1}{\frac{1}{f_1} + \frac{1}{f_2}} = \frac{f_1 f_2}{f_1 + f_2} \). Thus, the effective focal length of the lens system is \( f = \frac{f_1 f_2}{f_1 + f_2} \).

Key Concepts

Focal Length

Focal length is a fundamental concept in optics, describing the distance between the lens and the point where it focuses parallel incoming light rays. Understanding this concept is crucial when dealing with lens-based systems. In practical terms, the focal length tells you how strongly a lens converges or diverges light. Shorter focal lengths imply more powerful lenses that bend light significantly, while longer focal lengths mean less bending.

This concept is not just applicable to a single lens. When lenses are combined, as in the situation where two lenses are in contact, the individual focal lengths interact to create an effective focal length for the system. This is described by the formula:

• For two lenses in contact: \( f = \frac{f_1 f_2}{f_1 + f_2} \)

This formula signifies how the combined effect of two lenses can be captured in a single effective focal length. It highlights the fact that the effective focal length is always less than that of any individual lens in the pair, assuming both are positive.

Lens Combination

Combining lenses can create powerful optical systems. When two lenses are placed in contact, their combined effect is highly predictable using lens formulas. The combination of two lenses essentially means that each lens affects the light in sequence, and their individual powers add up.

• The power of a lens is simply the reciprocal of its focal length: \( |P| = \frac{1}{f} \).
• Therefore, in a system with two lenses in contact, the total power \( P \) is the sum of the two powers: \( P = |P_1| + |P_2| = \frac{1}{f_1} + \frac{1}{f_2} \).

These properties allow us to design complex optical devices, from straightforward glasses to intricate camera lenses, by carefully selecting the focal lengths to achieve desired focusing characteristics. Understanding how lens combination works enables the construction of solutions tailored to specific needs without having to use a single, highly complex lens.

Optics

Optics is the branch of physics concerned with the study of light and its interactions with lenses, mirrors, and other optical instruments. It reveals how light behaves as it travels through different mediums. The core goal is to understand how light can be manipulated to create images and improve vision, whether through the natural lens of the eye or artificial lenses.In the realm of thin lenses, which are lenses with negligible thickness compared to their radius of curvature, several important principles apply:

• Lenses can bend light rays to converge or diverge them, depending on their shape (concave or convex).
• Optical systems involving lenses rely on principles like refraction and the thin lens formula to determine image characteristics.
• The thin lens formula, \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), helps predict where an image will form and what its size will be compared to the object.

Mastering these concepts in optics allows us to harness the behavior of light to create a myriad of tools and technologies, from corrective eyewear to sophisticated imaging systems.