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Chapter 34: Problem 103

SSM Two thin lenses of focal lengths \(f_{1}\) and \(f_{2}\) are in contact. Show that 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 combined focal length is \( f = \frac{f_1 f_2}{f_1 + f_2} \).

Step by step solution

01

Understand the Lens Formula

The focal length of a lens can be related to the object and image distances using the lens formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance.
02

Define Individual Lens Contributions

For two thin lenses in contact, their combined optical effect can be found by adding their powers. The power of a lens is defined by \( P = \frac{1}{f} \). Therefore, for two lenses in contact, their combined power **P** is \( P = \frac{1}{f_1} + \frac{1}{f_2} \).
03

Derive the Combined Focal Length Formula

The combined power \( P \) is for a single lens equivalent to the two combined lenses. Thus, the focal length \( f \) of this equivalent lens is given by the power formula \( \frac{1}{f} = P = \frac{1}{f_1} + \frac{1}{f_2} \).Rearranging, we get:\[\frac{1}{f} = \frac{f_1 + f_2}{f_1 f_2}\]
04

Simplify the Combined Focal Length Formula

By rearranging the equation from Step 3, we can derive:\[f = \frac{f_1 f_2}{f_1 + f_2}\]This is the expression for the focal length of a single lens equivalent to the two lenses in contact.

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Key Concepts

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

Thin Lenses
In optics, a thin lens is defined as a lens with a thickness that is much smaller than its focal length and its radius of curvature. Thin lenses are idealized versions of real-world lenses, simplifying calculations and understanding. These lenses are used to focus or diverge light and are classified based on their shape and the refraction of light through them:
  • Convex Lenses: These lenses bulge outwards and converge light, focusing it to a point behind the lens. They have a positive focal length.
  • Concave Lenses: These lenses curve inwards and cause light rays to diverge. They have a negative focal length.
Real-life applications, such as cameras, glasses, and microscopes, often utilize thin lenses to manipulate light effectively, producing clear images for various optical instruments.
Focal Length Calculation
Focal length is a crucial concept in optics, determining how lenses refract light. Calculating the focal length involves understanding how it relates to the dimensions and characteristics of the lens, and it is often given by a formula based on the curvature of the lens surfaces and the index of refraction.When multiple lenses are used together, their combined focal length can have a significant impact.
  • If two lenses are in contact, their individual effects sum up to form a single combined optical power.
  • The combined focal length can be derived from the equation:\[f = \frac{f_1 f_2}{f_1 + f_2}\]This equation shows that the combined focal length is inversely proportional to the sum of the reciprocals of the individual focal lengths.
Using this formula, you can predict how a system of lenses will behave by understanding their individual focal lengths and how they work together.
Lens Formula
The lens formula is a foundational equation in optics, allowing us to relate the focal length of a lens to the distances of the object and image from the lens. It is expressed as:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]where:
  • \(f\) is the focal length of the lens.
  • \(d_o\) is the distance from the object to the lens.
  • \(d_i\) is the distance from the image to the lens.
This formula is used extensively in designing and analyzing optical systems. By understanding the relationship between these elements, one can determine how a lens will form an image of an object.For systems involving multiple lenses, the concept of power鈥攔eciprocal to the focal length鈥攃omes into play, allowing us to determine the effect of combining lenses. The simplicity and reliability of the lens formula make it a cornerstone of optical calculations, offering insight into how lenses modify light paths to form images.

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Most popular questions from this chapter

A lens is made of glass having an index of refraction of \(1.5\). One side of the lens is flat, and the other is convex with a radius of curvature of \(20 \mathrm{~cm} .\) (a) Find the focal length of the lens. (b) If an object is placed \(40 \mathrm{~cm}\) in front of the lens, where is the image?

ILW A moth at about eye level is \(10 \mathrm{~cm}\) in front of a plane mirror; you are behind the moth, \(30 \mathrm{~cm}\) from the mirror. What is the distance between your eyes and the apparent position of the moth's image in the mirror?

(a) 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 \(v_{I}\) 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). \(\begin{array}{ll}\text { sec, } 34 \cdot 6 & \text { Spherical Refracting Surfaces }\end{array}\)

. You look through a camera toward an image of a hummingbird in a plane mirror. The camera is \(4.30 \mathrm{~m}\) in front of the mirror. The bird is at camera level, \(5.00 \mathrm{~m}\) to your right and \(3.30 \mathrm{~m}\) from the mirror. What is the distance between the camera and the apparent position of the bird's image in the mirror?

ssin A fruit fly of height \(H\) sits in front of lens 1 on the central axis through the lens. The lens forms an image of the fly at a distance \(d=20 \mathrm{~cm}\) from the fly; the image has the fly's orientation and height \(H_{l}=2.0 H .\) What are (a) the focal length \(f_{1}\) of the lens and (b) the object distance \(p_{1}\) of the fly? The fly then leaves lens
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