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Chapter 27: Problem 93

A Big Eye The largest eye ever to exist on Earth belonged to an extinct species of ichthyosaur, Temmodontosaurus platyodon. This creature had an eye that was \(26.4 \mathrm{cm}\) in diame ter. It is estimated that this ichthyosaur also had a relatively large pupil, giving it an effective aperture setting of about \(f / 1.1\). (a) Assuming its pupil was one-third the diameter of the eye, what was the approximate focal length of the ichthyosaur's eye? (b) When the ichthyosaur narrowed its pupil in bright light, did its \(f\) -number increase or decrease? Explain.

Short Answer

Expert verified
(a) The focal length is 9.68 cm. (b) The f-number increases when the pupil narrows in bright light.

Step by step solution

01

Understand the f-number formula

The f-number (or f-stop, denoted as \(f/\)) is given by the ratio of the focal length of the lens \(f\) to the diameter of the entrance pupil (D). The formula is: \[f\text{-number} = \frac{f}{D}\]
02

Calculate the diameter of the pupil

We're given that the pupil is one-third the diameter of the eye, which has a diameter of 26.4 cm. Therefore, the pupil's diameter \(D\) is: \[D = \frac{26.4}{3} = 8.8 \text{ cm}\]
03

Solve for the focal length

We know the effective f-number is 1.1, and we've calculated the pupil's diameter to be 8.8 cm. Using the f-number formula: \[1.1 = \frac{f}{8.8}\] Multiply both sides by 8.8 to solve for \(f\):\[f = 1.1 \times 8.8 = 9.68 \text{ cm}\]
04

Analyze the effect of pupil narrowing on the f-number

The f-number is inversely proportional to the diameter of the pupil. When the pupil narrows in bright light, the diameter decreases, which increases the f-number according to the formula: \(f/ \) = \(\frac{f}{D}\).

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Key Concepts

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

F-Number Formula
In optics, the f-number (or f-stop) is an important concept that describes the amount of light a lens can gather. It's a dimensionless number that represents the ratio of the focal length of a lens to the diameter of its entrance pupil. In mathematical terms, this is expressed as: \[ f\text{-number} = \frac{f}{D} \] where
  • \( f \) represents the focal length of the lens.
  • \( D \) stands for the diameter of the entrance pupil.
A smaller f-number means a larger aperture, allowing more light to enter, which is helpful in low-light conditions. Conversely, a larger f-number implies a smaller aperture, allowing less light.
Pupil Diameter
The pupil diameter is crucial in determining the f-number because it directly affects the amount of light entering the eye or lens. In the case of the ichthyosaur, the pupil is one-third the diameter of its large eye, which was 26.4 cm. Calculating the pupil diameter involves dividing the eye's diameter by three:
  • \( D = \frac{26.4\text{ cm}}{3} = 8.8 \text{ cm} \)
This measurement signifies the entrance size for light, impacting how much light reaches the retina or sensor. The larger the pupil diameter, the more light it can collect, affecting the overall image brightness. Keep in mind, as light conditions change, so might the pupil diameter, altering the f-number.
Focal Length Calculation
To find the focal length using the f-number formula, one rearranges the equation to solve for \( f \) when given \( D \) and the effective f-number. The focal length is a measurement of how strongly the optical system converges or diverges light. For the ichthyosaur's eye, given
  • \( f\text{-number} = 1.1 \)
  • \( D = 8.8 \text{ cm} \)
the focal length \( f \) can be calculated by multiplying the f-number by the pupil diameter:\[ f = 1.1 \times 8.8 = 9.68 \text{ cm} \]Thus, the focal length of the ichthyosaur's eye was approximately 9.68 cm, indicating the distance over which the light rays are brought to a focus.
Aperture Setting
The aperture setting refers to the size of the opening through which light enters a lens or eye, affecting the exposure and depth of field in photography or perception in animals. The ichthyosaur had an effective aperture setting of \( f/1.1 \), meaning it had a relatively large aperture capable of gathering a lot of light quickly. When an aperture setting like this is used, it typically results in
  • Brighter images in dim conditions.
  • A shallow depth of field.
However, when the pupil narrows, such as in bright environments to protect sensitive retinal cells, the aperture decreases, increasing the f-number and reducing the amount of light collected. Understanding these dynamics helps in adjusting both visual and photographic exposure effectively.

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Most popular questions from this chapter

The lens in a normal human eye, with aqueous humor on one side and vitreous humor on the other side, has a a refractive power of 15 diopters. Suppose a lens is removed from an eye and surrounded by air. In this case, is its refractive power greater than, less than, or equal to 15 diopters? Explain.

To construct a telescope, you are given a lens with a focal length of \(32 \mathrm{mm}\) and a lens with a focal length of \(1600 \mathrm{mm}\). (a) On the basis of focal length alone, which lens should be the objective and which the eyepiece? Explain. (b) What magnification would this telescope produce?

The focal length of a relaxed human eye is approximately \(1.7 \mathrm{cm}\). When we focus our eyes on a close-up object, we can change the refractive power of the eye by about 16 diopters. (a) Does the refractive power of our eyes increase or decrease by 16 diopters when we focus closely? Explain. (b) Calculate the focal length of the eye when we focus closely.

A patient receives a rigid IOL whose focus cannot be changed-it is designed to provide clear vision of objects at infinity. The patient will use corrective contacts to allow for close vision. Should the refractive power of the corrective contacts be positive or negative?

Because a concave lens cannot form a real image of a real object, it is difficult to measure its focal length precisely. One method uses a second, convex, lens to produce a virtual object for the concave lens. Under the proper conditions, the concave lens will form a real image of the virtual object! \(A\) student conducting a laboratory project on concave lenses makes the following observations: When a lamp is placed \(42.0 \mathrm{cm}\) to the left of a particular convex lens, a real (inverted) image is formed \(37.5 \mathrm{cm}\) to the right of the lens. The lamp and convex lens are kept in place while a concave lens is mounted \(15.0 \mathrm{cm}\) to the right of the convex lens. A real image of the lamp is now formed \(35.0 \mathrm{cm}\) to the right of the concave lens. What is the focal length of each lens?
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