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The focal length of an \(f / 4\) camera lens is \(300 \mathrm{~mm}\). (a) What is the aperture diameter of the lens? (b) If the correct exposure of a certain scene is \(\frac{1}{250} \mathrm{~s}\) at \(f / 4,\) what is the correct exposure at \(f / 8 ?\)

Short Answer

Expert verified
(a) 75 mm, (b) \(\frac{1}{60}\) s.

Step by step solution

01

Identify the Aperture Diameter Formula

The aperture diameter is given by the formula \( D = \frac{f}{N} \), where \( D \) is the aperture diameter, \( f \) is the focal length, and \( N \) is the f-number or f-stop.
02

Calculate for f/4

For the given f-stop \( f/4 \), the focal length \( f \) is \( 300 \text{ mm} \). To find the aperture diameter, substitute into the formula: \( D = \frac{300 \text{ mm}}{4} = 75 \text{ mm} \).
03

Understanding the Relationship Between F-stops and Exposure

The exposure changes when the f-stop changes are calculated by knowing that reducing the aperture by one stop (e.g., from \( f/4 \) to \( f/8 \)) reduces the amount of light by a factor of four.
04

Calculate Exposure Time at f/8

To maintain the same exposure when moving from \( f/4 \) to \( f/8 \), the exposure time must be increased. The exposure time at \( f/8 \) is calculated as \( 4 \times \frac{1}{250} = \frac{4}{250} = \frac{1}{62.5} \text{ s} \), which can be approximated to \( \frac{1}{60} \text{ s} \).

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Key Concepts

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

Focal Length
When we talk about focal length, we're discussing a crucial element in the world of optics. Focal length is the distance between the lens and the image sensor when the subject is in focus. It is measured in millimeters (mm). For example, in our exercise, the focal length is specified as \(300 \text{ mm}\).
Focal length affects how "zoomed in" your photos are. A longer focal length means a higher magnification, allowing you to view distant subjects larger in scale. Conversely, a shorter focal length provides a wider view, suitable for capturing more landscape.
Here's what to keep in mind:
  • Long focal lengths: Larger subjects, narrow field of view.
  • Short focal lengths: Smaller subjects, wide field of view.
Understanding focal length helps in choosing the right lens for your photography needs, impacting how your subjects appear on film.
Aperture Diameter
The aperture diameter is the hole within a lens through which light travels. It can be thought of as the 'eye' of your camera. A larger diameter allows more light in, crucial for taking clear shots in low-light situations. In our exercise, the formula \(D = \frac{f}{N}\) is used to find this diameter, where \(f\) stands for the focal length, and \(N\) represents the f-stop.
For instance, with a focal length of \(300 \text{ mm}\) and an f-stop of \(4\), the aperture diameter is \(75 \text{ mm}\). This calculation is essential for photographers who need to adjust the light intake on their camera.
Remember these key points:
  • Larger diameter: More light, better for dark scenes.
  • Smaller diameter: Less light, stronger depth of field.
Understanding aperture diameter helps in controlling exposure and the aesthetic quality of your photos.
F-stop
f-stop or f-number is a term used to describe the size of the aperture hole relative to the focal length. It is denoted as \(f/N\), where \(N\) is the actual f-stop number. In our exercise, \(f/4\) is mentioned, and adjusting to \(f/8\) affects how the image is exposed.
Here's a breakdown:
  • Lower f-stop (e.g., f/1.8, f/2.8): Larger aperture, more light, shallow depth of field.
  • Higher f-stop (e.g., f/8, f/16): Smaller aperture, less light, greater depth of field.
The f-stop scale follows a geometric sequence, where each "stop" halves or doubles the light. For instance, moving from \(f/4\) to \(f/8\) reduces the light by one fourth. It's critical for precise control over exposure and depth of field.
Exposure Time
Exposure time, or shutter speed, is the duration for which the camera's sensor is exposed to light. It's pivotal in capturing the right level of detail and light in a photo. In our task, starting at an exposure of \(\frac{1}{250} \text{ s}\) at \(f/4\), adjusting to \(f/8\) requires an increased exposure time of \(\frac{1}{60} \text{ s}\).
Exposure time influences motion blur and image brightness:
  • Short exposure time (e.g., \(\frac{1}{1000} \text{ s}\)): Freezes motion but less light.
  • Long exposure time (e.g., \(1 \text{ s}\)): More light, but can blur moving subjects.
Finding the right balance between a clear image and achieving the desired lighting is vital. By understanding exposure time, photographers can better adjust settings to adapt to different lighting conditions and desired artistic effects.

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Most popular questions from this chapter

A \(135 \mathrm{~mm}\) telephoto lens for a \(35 \mathrm{~mm}\) camera has \(f\) -stops that range from \(f / 2.8\) to \(f / 22 .\) (a) What are the smallest and largest aperture diameters for this lens? What is the diameter at \(f / 11 ?\) (b) If a \(50 \mathrm{~mm}\) lens had the same \(f\) -stops as the telephoto lens, what would be the smallest and largest aperture diameters for that lens? (c) At a given shutter speed, what is the ratio of the greatest to the smallest light intensity of the film image? (d) If the shutter speed for correct exposure at \(f / 22\) is \(1 / 30 \mathrm{~s},\) what shutter speed is needed at \(f / 2.8 ?\) Calculate \(m_{1}\) and \(M_{2}\) for the two lenses and do not make the approximation that leads to Equation 25.4

An amateur photographer purchases a vintage camera at a flea market. In order to determine the focal length of the camera's lens, he sets a soup can at various distances from the camera. At each distance he brings the camera into focus on the can and then carefully measures the distance between the lens and the film. His measurements are given in the table. $$ \begin{array}{cc} \hline \text { Can distance }(\mathrm{cm}) & \text { Lens distance }(\mathrm{mm}) \\ \hline 25 & 118 \\ 35 & 104 \\ 50 & 95 \\ 75 & 89 \\ 100 & 87 \\ \hline \end{array} $$ Make a plot of the inverse lens distance as a function of the inverse can distance. Using a linear "best fit" to the data, determine the focal length of the lens.

An insect \(1.2 \mathrm{~mm}\) tall is placed \(1.0 \mathrm{~mm}\) beyond the focal point of the objective lens of a compound microscope. The objective lens has a focal length of \(12 \mathrm{~mm}\), the eyepiece a focal length of \(25 \mathrm{~mm} .\) (a) Where is the image formed by the objective lens, and how tall is it? (b) If you want to place the eyepiece so that the image it produces is at infinity, how far should this lens be from the image produced by the objective lens? (c) Under the conditions of part (b), find the overall magnification of the microscope.

A student's far point is at \(17.0 \mathrm{~cm},\) and she needs glasses to view her computer screen comfortably at a distance of \(45.0 \mathrm{~cm}\). What should be the power of the lenses for her glasses?

BIO A person can see clearly up close but cannot focus on objects beyond \(75.0 \mathrm{~cm} .\) She opts for contact lenses to correct her vision. (a) Is she nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed to correct her vision? (c) What focallength contact lens is needed, and what is its power in diopters?

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