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Chapter 34: Problem 103

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 \(f=f_{1} f_{2} /\left(f_{1}+f_{2}\right)\)

Short Answer

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

Step by step solution

01

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.
02

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 \( P_2 = \frac{1}{f_2} \).
03

Calculating the Combined Focal Length

Substitute the powers \( P_1 \) and \( P_2 \) into the equation for combined power: \( P = \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \).
04

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 \( f = \frac{f_1 f_2}{f_1 + f_2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Two-lens systems. In Fig. \(34-45\), stick figure \(O\) (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to \(O\), which is at object distance \(p_{1}\). Lens 2 is mounted within the farther boxed region, at distance \(d .\) Each problem in Table \(34-9\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(\mathrm{C}\) for converging and D for diverging; the number after C or \(\mathrm{D}\) is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 2 as object \(O\) or on the opposite side. \(\begin{array}{lllll}\mathbf{8 0} & +10 & \mathrm{C}, 15 & 10 & \mathrm{C}, 8.0\end{array}\)

The formula \(1 / p+1 / i=1 / f\) is called the Gaussian form of the thin-lens formula. Another form of this formula, the Newtonian form, is obtained by considering the distance \(x\) from the object to the first focal point and the distance \(x^{\prime}\) from the second focal point to the image. Show that \(x x^{\prime}=f^{2}\) is the Newtonian form of the thin-lens formula.

A luminous point is moving at speed \(v_{o}\) toward a spherical mirror with radius of curvature \(r\), along the central axis of the mirror. Show that the image of this point is moving at speed $$ v_{I}=-\left(\frac{r}{2 p-r}\right)^{2} v_{O} $$ where \(p\) is the distance of the luminous point from the mirror at any given time. Now assume the mirror is concave, with \(r=15 \mathrm{~cm}\), and let \(v_{O}=5.0 \mathrm{~cm} / \mathrm{s}\). Find \(v_{l}\) when (b) \(p=30 \mathrm{~cm}\) (far outside the focal point), (c) \(p=8.0 \mathrm{~cm}\) (just outside the focal point), and (d) \(p=10 \mathrm{~mm}\) (very near the mirror).

stick figure \(O\) (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to \(O\), which is at object distance \(p_{1} .\) Lens 2 is mounted within the middle boxed region, at distance \(d_{12}\) from lens \(1 .\) Lens 3 is mounted in the farthest boxed region, at distance \(d_{23}\) from lens \(2 .\) Each problem in Table \(34-10\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(C\) for converging and \(D\) for diverging: the number after \(\mathrm{C}\) or \(\mathrm{D}\) is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 3 as object \(O\) or on the opposite side. \begin{tabular}{lllllll} \(\mathbf{1 0 0}\) & \(+4.0\) & \(\mathbf{C}, 6.0\) & \(8.0\) & \(\mathrm{D}, 4.0\) & \(5.7\) & \(\mathrm{D}, 12\) \\ \hline \end{tabular}

A concave shaving mirror has a radius of curvature of \(35.0 \mathrm{~cm}\). It is positioned so that the (upright) image of a man's face is \(2.50\) times the size of the face. How far is the mirror from the face?
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