Problem 4 - College Physics

Question

. A 135 mm telephoto lens for a 35 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\(/ 30\) s, what shutter speed is needed at \(f / 2.8 ?\)

Step-by-Step Solution

Answer

(a) 6.14 mm to 48.21 mm, 12.27 mm (b) 2.27 mm to 17.86 mm (c) 62.03 (d) 0.5 s

Step 1: Understand the Lens Equation

The aperture diameter is related to the focal length and the f-stop number by the equation: \( D = \frac{f}{N} \), where \( D \) is the aperture diameter, \( f \) is the focal length, and \( N \) is the f-stop number.

Step 2: Calculate Smallest and Largest Apertures for 135 mm Lens

For the smallest aperture at \( f/22 \): \( D = \frac{135}{22} \approx 6.14 \) mm. For the largest aperture at \( f/2.8 \): \( D = \frac{135}{2.8} \approx 48.21 \) mm.

Step 3: Calculate Diameter at f/11 for 135 mm Lens

For \( f/11 \), \( D = \frac{135}{11} \approx 12.27 \) mm.

Step 4: Calculate Smallest and Largest Apertures for 50 mm Lens

For the smallest aperture at \( f/22 \): \( D = \frac{50}{22} \approx 2.27 \) mm. For the largest aperture at \( f/2.8 \): \( D = \frac{50}{2.8} \approx 17.86 \) mm.

Step 5: Determine Light Intensity Ratio

The light intensity is proportional to the area of the aperture, which is \( \pi \left(\frac{D}{2}\right)^2 \). The intensity ratio \( I \) from \( f/2.8 \) to \( f/22 \) is \( \left(\frac{22}{2.8}\right)^2 \approx 62.03 \).

Step 6: Calculate Shutter Speeds at Different f-Stops

The exposure for correct exposure, \( E \), is constant: \( E = \text{intensity} \times \text{time} = \text{constant} \). For \( f/22 \) at 1/30 s, the shutter speed \( \text{t} \) needed at \( f/2.8 \) can be calculated by \( 30 \div 62.03 \approx 0.484 \approx 0.5 \) seconds.

Key Concepts

f-stop

The f-stop, commonly denoted by \( f/N \), is a crucial concept in photography, defining the size of the lens aperture. It is essentially the ratio of the lens's focal length \( f \) to the diameter \( D \) of the aperture:

• F-stop is essential for controlling the depth of field and exposure in a photograph.
• Lower f-stop numbers (e.g., \( f/2.8 \)) result in larger aperture diameters, allowing more light to hit the camera sensor.
• Higher f-stop numbers (e.g., \( f/22 \)) correspond to smaller aperture diameters, resulting in less light and a greater depth of field.

The f-stop number indicates a geometric sequence where each full stop either halves or doubles the amount of light entering the lens compared to the adjacent stop.

focal length

Focal length is a measure of how strongly a lens converges or diverges light. In practical terms, it defines the lens's "zoom" capability and affects the field of view of the image.

• A longer focal length (e.g., 135 mm in our example) gives a narrower field of view and reduces the amount of the scene captured.
• Shorter focal lengths (e.g., 50 mm) capture a wider angle of the scene.
• The focal length also influences the perspective and the scale of subjects in the photograph; distant subjects appear closer with a longer focal length.

It is important for photographers to understand how changing the focal length impacts the composition and proportions in their images.

light intensity

Light intensity on the film or sensor of a camera is crucial for ensuring the image is neither too bright nor too dark. It is influenced by the aperture size, as the aperture determines how much light reaches the sensor.

• The relationship is such that the area of the aperture controls light intensity: larger apertures (lower f-stops) admit more light.
• The intensity varies greatly across f-stops; for instance, moving from \( f/2.8 \) to \( f/22 \), the ratio of light intensity is significant, as calculated by the ratio formula \( I = \left(\frac{22}{2.8}\right)^2 \).
• This significant difference in intensity is why understanding f-stops is vital for correct exposure in varying lighting conditions.

shutter speed

Shutter speed is the duration for which the camera's sensor is exposed to light, greatly influencing the exposure and motion blur in photography.

• A faster shutter speed freezes action and is useful in bright conditions, while a slower speed may create motion blur, useful for artistic effects.
• When changing f-stops, adjusting the shutter speed is necessary to maintain correct exposure if the light intensity changes.
• For example, when the aperture is set to \( f/22 \) with a speed of 1/30 s, switching to \( f/2.8 \) requires compensating with a slower shutter speed, approximately 0.5 s, to ensure consistent exposure.

Balancing aperture and shutter speed is key to creating well-exposed and visually compelling images.