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Figure P34.99 shows a simple version of a zoom lens. The converging lens has focal length \(f_{1}\) and the diverging lens has focal length \(f_{2}=-\left|f_{2}\right| .\) The two lenses are separated by a variable distance \(d\) that is always less than \(f_{1}\). Also, the magnitude of the focal length of the diverging lens satisfies the inequality \(\left|f_{2}\right|>\left(f_{1}-d\right) .\) To determine the effective focal length of the combination lens, consider a bundle of parallel rays of radius \(r_{0}\) entering the converging lens. (a) Show that the radius of the ray bundle decreases to \(r_{0}^{\prime}=r_{0}\left(f_{1}-d\right) / f_{1}\) at the point that it enters the diverging lens. (b) Show that the final image \(I^{\prime}\) is formed a distance \(s_{2}^{\prime}=\left|f_{2}\right|\left(f_{1}-d\right) /\left(\left|f_{2}\right|-f_{1}+d\right)\) to the right of the diverging lens. (c) If the rays that emerge from the diverging lens and reach the final image point are extended backward to the left of the diverging lens, they will eventually expand to the original radius \(r_{0}\) at some point \(Q .\) The distance from the final image \(I^{\prime}\) to the point \(Q\) is the effective focal length \(f\) of the lens combination; if the combination were replaced by a single lens of focal length \(f\) placed at \(Q\), parallel rays would still be brought to a focus at \(I^{\prime} .\) Show that the effective focal length is given by \(f=f_{1}\left|f_{2}\right| /\left(\left|f_{2}\right|-f_{1}+d\right) .\) (d) If \(f_{1}=12.0 \mathrm{~cm}, f_{2}=-18.0 \mathrm{~cm},\) and the separation \(d\) is adjustable between 0 and \(4.0 \mathrm{~cm}\), find the maximum and minimum focal lengths of the combination. What value of \(d\) gives \(f=30.0 \mathrm{~cm} ?\)

Step by step solution

01

Formula for radius of the ray bundle

Given that the radius of the ray bundle decreases to \(r_{0}^{\prime}\) at the point that it enters the diverging lens, we can express this mathematically as: \(r_{0}^{\prime}=r_{0}\left(f_{1}-d\right) / f_{1}\). This shows the relationship between the radius of the ray bundle before and after it reaches the diverging lens.

02

Formula for the distance where the final image is formed

To find the position where the final image is formed, we express the distance as \(s_{2}^{\prime}=\left|f_{2}\right|\left(f_{1}-d\right) /\left(\left|f_{2}\right|-f_{1}+d\right)\). We assume that the light travels from left to right and 'to the right' refers to the direction of light propagation from the lens.

03

Formula for the effective focal length

The effective focal length of the lens combination is defined as \(f=f_{1}\left|f_{2}\right| /\left(\left|f_{2}\right|-f_{1}+d\right)\). This combines both focal lengths of the individual lenses and their separation.

04

Calculate the maximum and minimum focal lengths

For \(f_{1}=12.0 \mathrm{~cm}\), \(f_{2}=-18.0 \mathrm{~cm}\), and the separation \(d\) is adjustable between 0 and \(4.0 \mathrm{~cm}\), find the maximum and minimum focal lengths by plugging in the values of \(d\) in the expression for \(f\) at the boundaries.

05

Calculate the value of \(d\) when \(f=30.0 \mathrm{~cm}\)

Finally, we determine the separation \(d\) that makes the effective focal length to be \(30.0 \mathrm{~cm}\). We can accomplish this by setting \(f=30.0 \mathrm{~cm}\) in the expression for \(f\) and solving for \(d\).

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Key Concepts

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

Converging and Diverging Lenses

The basics of understanding a zoom lens system lie in the roles of converging and diverging lenses. A converging lens, often called a convex lens, has a thicker center compared to its edges, allowing it to bend incoming parallel light rays towards a common point on the principal axis. This point is known as the focal point. Converging lenses are characterized by positive focal lengths, meaning they bring light together or "converge" it.

On the other hand, a diverging lens, also known as a concave lens, displays the opposite behavior. It is thinner in the middle and causes parallel incoming light rays to spread out, or "diverge". Diverging lenses have negative focal lengths, indicating their tendency to disperse light rather than concentrate it. In a typical zoom lens, both types of lenses work together, with the converging lens initially focusing the light and the diverging lens then altering the path of rays to adjust the effective focal length.

• **Converging lenses** focus light to a point, aiding in magnification.
• **Diverging lenses** spread light, useful for wide-angle views.

Understanding how these lenses work in tandem is essential for mastering the concept of a variable focal length in zoom lenses.

Focal Length

Focal length is a critical parameter in determining how a lens will focus light. It is the distance from the lens to the point where it converges parallel light rays, known as the focal point, for converging lenses, or imaginary for diverging lenses. The focal length is measured in centimeters or millimeters and can convey much about the lens’s properties and potential applications. For converging lenses, the focal length is positive, whereas, for diverging lenses, it is negative.

In a zoom lens system, adjusting the focal length involves varying the distance between the lenses. This results in changes to the lens's effective focal length, which manipulates the magnification and field of view. Therefore, knowing how to calculate the effective focal length through the formula: \[ f = \frac{f_1 |f_2|}{|f_2| - f_1 + d} \] allows one to determine how lens adjustments will alter image formation.

• The focal length dictates the lens's magnifying power.
• Adjusting the focal length changes the zoom level.

By understanding this concept, students can grasp how simple lens setups can lead to complex image manipulations.

Optical Lenses

Optical lenses play a crucial role in focusing or dispersing light, which fundamentally governs how images are formed in photographic equipment, glasses, and telescopes. The key function of optical lenses is to manage light paths. They influence how light travels through mediums with different refractive indices, precisely redirecting the light rays to form an image at the desired location.

There are several types of optical lenses, yet in the context of a zoom lens, we most often deal with converging and diverging types. These are utilized to alter the path of light rays to achieve varied focal lengths, providing the essence of zoom functionality.

• **Converging lenses**: Direct light to a point, useful in cameras and other imaging systems.
• **Diverging lenses**: Spread light out, common in certain corrective lenses and projectors.

Comprehending how optical lenses manipulate light provides insights into achieving different focal lengths and enhances one's ability to apply these concepts to practical scenarios, such as creating sharper images or changing perspectives in photography.