Problem 1 - College Physics

Question

\(\cdot\) 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 \(\frac{1}{250}\) s at \(f / 4,\) what is the correct exposure at \(f / 8 ?\)

Step-by-Step Solution

Answer

(a) 75 mm; (b) \(\frac{1}{62.5}\) s.

Step 1: Understanding the Given Information

We start with an \(f/4\) camera lens which has a focal length of 300 mm. The \(f\)-number describes the aperture size in relation to the focal length. The correct exposure at \(f/4\) is given as \(\frac{1}{250}\) s, and we need to calculate the exposure time for \(f/8\).

Step 2: Calculating the Aperture Diameter

The formula for the aperture diameter \(D\) in relation to the \(f\)-number \(N\) and the focal length \(f\) is \(D = \frac{f}{N}\). For \(f/4\), the aperture diameter is \(D = \frac{300\,\text{mm}}{4} = 75\,\text{mm}\).

Step 3: Understand \(f/\) Number Changes

When the \(f\)-number changes from \(f/4\) to \(f/8\), the amount of light reaching the sensor decreases. Since \(f/8\) is two \(f\)-stops smaller than \(f/4\), the light is reduced by a factor of 4.

Step 4: Calculating the Correct Exposure Time for \(f/8\)

To compensate for the reduced light at \(f/8\), we need to increase the exposure time by a factor of 4. The original exposure time at \(f/4\) is \(\frac{1}{250}\) s, so the new exposure time at \(f/8\) is \(4 \times \frac{1}{250} = \frac{1}{62.5}\) s.

Key Concepts

Aperture Diameter

The term "aperture diameter" refers to the opening through which light enters a camera lens. This opening plays a crucial role in determining the amount of light that reaches the camera's sensor. A larger aperture diameter lets in more light, which can be particularly useful in low-light conditions. The aperture diameter can be calculated using the formula:
\[ D = \frac{f}{N} \]where:

• \(D\) is the aperture diameter.
• \(f\) is the focal length of the lens.
• \(N\) is the \(f\)-number.

In our exercise, with a focal length of 300 mm and an \(f/4\) setting, the aperture diameter is 75 mm. This size allows ample light to penetrate the lens, making it suitable for a variety of lighting situations.

Exposure Time

Exposure time, also known as shutter speed, is the duration for which the camera's shutter is open to allow light to hit the sensor. It plays a vital role in photography, affecting the brightness and clarity of the photo. In simpler terms, a longer exposure time means more light is captured, resulting in a brighter image.In the problem provided, the correct exposure time at \(f/4\) is given as \(\frac{1}{250}\) seconds. When changing to \(f/8\), we need to adjust the exposure time to account for the reduced light. Since \(f/8\) lets in less light than \(f/4\), we need to increase the exposure time by a factor of 4 (as two \(f\)-stops reduce the light). Therefore, the adjusted exposure time for \(f/8\) is \(\frac{1}{62.5}\) seconds, which helps maintain the necessary light for proper image exposure.

f-stop

The f-stop, or \(f\)-number, is a measure of the lens's aperture size relative to its focal length. It controls the depth of field in a photo and influences how much light enters through the lens. The \(f\)-stop scale is such that:

• Lower \(f\)-numbers (e.g., \(f/2\), \(f/4\)) have larger aperture openings and allow more light.
• Higher \(f\)-numbers (e.g., \(f/8\), \(f/16\)) reduce the aperture size and allow less light.

Switching from \(f/4\) to \(f/8\) in the exercise reduces the light by a factor of 4. This is because each increase by one \(f\)-stop (e.g., from \(f/4\) to \(f/5.6\), then \(f/5.6\) to \(f/8\)) halves the amount of light reaching the sensor.