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A camera lens has a focal length of 180.0 \(\mathrm{mm}\) and an aperture diameter of 16.36 \(\mathrm{mm}\) . (a) What is the \(f\) -number of the lens? (b) If the correct exposure of a certain scene is \(\frac{1}{30} \mathrm{s}\) at \(f / 11\) what is the correct exposure at \(f / 2.8 ?\)

Short Answer

Expert verified
The f-number is approximately 11. The correct exposure at \( f/2.8 \) is \( \frac{1}{480} \) s.

Step by step solution

01

Understanding the f-number Formula

The f-number or f-stop is calculated using the formula \( f/N = \frac{F}{D} \), where \( F \) is the focal length of the lens, and \( D \) is the diameter of the aperture. In this problem, \( F = 180 \) mm and \( D = 16.36 \) mm.
02

Calculating the f-number

Substitute the given values into the f-number formula: \[ f/N = \frac{180}{16.36} \approx 11.0 \]. Thus, the f-number of the camera lens is approximately \( f/11 \).
03

Understanding Exposure and f-number

Exposure corresponds to how much light hits the camera sensor, calculated considering the f-number adjustment. At a higher f-number such as \( f/11 \), the aperture is smaller, allowing less light in compared to a lower f-number like \( f/2.8 \). Thus, with a lower f-number, shorter exposure times are needed as more light comes in.
04

Changing Exposure Based on f-number

Shutter speed must compensate for the f-number change to maintain correct exposure. Changing from \( f/11 \) to \( f/2.8 \) involves changing the light exposure by the ratio of the areas (\( \left(\frac{11}{2.8}\right)^2 \)). The square of \( \frac{11}{2.8} \approx 4 \), which indicates that \( 4^2 = 16 \) times more light is allowed at \( f/2.8 \).
05

Calculating Correct Exposure Time

Since changing from \( f/11 \) to \( f/2.8 \) permits 16 times more light, the exposure time should be 16 times shorter to compensate. At \( f/11 \), the exposure time was \( \frac{1}{30} \) s, so \( \frac{1}{30} \times \frac{1}{16} = \frac{1}{480} \) s. Hence, the correct exposure at \( f/2.8 \) is \( \frac{1}{480} \) s.

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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 crucial characteristic of a camera lens. It determines how much of a scene gets captured and how big objects appear in the image. Measured in millimeters, a longer focal length will zoom into a scene, capturing less but making subjects larger.

Think of it as how far your camera "sees." A telephoto lens with a long focal length, like 200mm, brings distant objects closer. Conversely, a shorter focal length, such as 18mm, is like seeing through a wide-angle lens, capturing more of the scene.

In our example, the focal length is 180mm. This means the lens has a moderate telephoto effect, ideal for portraits or photos requiring a slight zoom.
Aperture Diameter
The aperture diameter is essentially the size of the opening in a camera lens. This opening controls the amount of light that enters the camera. Measured in millimeters, a larger aperture lets more light through, which is great for low-light conditions.

In addition, aperture diameter affects the depth of field, or how much of your photograph is in focus. A wide aperture (low f-number) results in a shallow depth of field, perfect for isolating subjects. Meanwhile, a narrow aperture (high f-number) deepens the focus.

Our example had an aperture diameter of 16.36mm, calculated into an f-number of f/11. This setting balances light intake and scene focus, suitable for general photography.
Exposure Time
Exposure time, often referred to as shutter speed, is crucial to how your photographs turn out. It is the duration the camera's shutter remains open to let in light. Faster shutter speeds freeze action, making them ideal for sports or fast-moving events. Slower speeds allow more light in but can cause blurring if the camera or subject moves.

Measured in fractions of a second, like 1/30 s, altering exposure time compensates for changes in aperture size, ensuring images are correctly exposed. If a lens lets in more light (like when using a bigger aperture), a shorter exposure time is needed.

In our scenario, reducing the exposure time from 1/30 s to 1/480 s balances out the increased light from changing to a lower f-number.
Shutter Speed
Shutter speed is a critical aspect of photography that works hand in hand with aperture to create stunning images. It represents how quickly the camera's shutter opens and closes. This speed determines how motion is captured—whether it's a crisp snapshot or a flowing blur.

Fast shutter speeds, like 1/1000 s, are excellent for capturing fast actions like a bird in flight. In contrast, slow speeds, like 1/10 s, are used for night scapes or artistic blurs, as they allow more light and motion into the exposure.

Our exercise adjusted the shutter speed from 1/30 s, appropriate for a higher f-number, to 1/480 s. This change ensured that even with more light from a larger aperture, the photo remained well-exposed, maintaining the right balance of light and motion.

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Most popular questions from this chapter

A converging lens forms an image of an 8.00 -mm-tall real object. The image is 12.0 \(\mathrm{cm}\) to the left of the lens, 3.40 \(\mathrm{cm}\) tall, and erect. What is the focal length of the lens? Where is the object located?

Recall that the intensity of light reaching film in a camera is proportional to the effective area of the lens. Camera A has a lens with an aperture diameter of 8.00 \(\mathrm{mm} .\) It photographs an object using the correct exposure time of \(\frac{1}{30} \mathrm{s}\) . What exposure time should be used with camera \(\mathrm{B}\) in photographing the same object with the same film if this camera has a lens with an aperture diameter of 23.1 \(\mathrm{mm}\) ?

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 focal length contact lens is needed, and what is its power in diopters?

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