Problem 38 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Human vision normally covers an angle of about \(40^{\circ}\) horizontally. A "normal" camera lens then is defined as follows: When focused on a distant horizontal object which subtends an angle of \(40^{\circ}\), the lens produces an image that extends across the full horizontal extent of the camera's light-recording medium (film or electronic sensor). Determine the focal length \(f\) of the "normal" lens for the following types of cameras: \((a)\) a \(35-\mathrm{mm}\) camera that records images on film \(36 \mathrm{~mm}\) wide; \((b)\) a digital camera that records images on a charge-coupled device \((\mathrm{CCD}) 1.00 \mathrm{~cm}\) wide.

Step-by-Step Solution

Answer

(a) approx. 49.45 mm, (b) approx. 13.74 mm.

Step 1: Convert Angle to Radians

The angle of the field of view given is \( 40^\circ \). To perform calculations using trigonometric functions, we first convert this angle to radians using the formula: \[\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\]So, the angle in radians is \[40^\circ = \frac{\pi}{180} \times 40 = \frac{2\pi}{9} \text{ radians}\]

Step 2: Understand the Lens Equation

The lens focuses an image such that an object subtending an angle \(\theta\) produces an image of width equal to the sensor/film width. This forms a right triangle with the relationship:\[\tan\left(\frac{\theta}{2}\right) = \frac{\text{half width of sensor}}{f}\]This equation will help us find the focal length \( f \).

Step 3: Calculate Focal Length for 35-mm Camera

The width of the film is \( 36 \text{ mm} \). Thus, half the width is \( 18 \text{ mm} \). Use the formula:\[\tan\left(\frac{20^\circ}{2}\right) = \frac{18}{f}\]Calculate \( f \):\[\tan\left(20^\circ\right) = 0.364\]\[f = \frac{18}{0.364} \approx 49.45 \text{ mm}\]

Step 4: Calculate Focal Length for Digital Camera

The width of the CCD is \( 1.00 \text{ cm} \), which is \( 10 \text{ mm} \). Hence, half the width is \( 5 \text{ mm} \). The calculation is similar:\[\tan\left(20^\circ\right) = \frac{5}{f}\]Calculate \( f \):\[f = \frac{5}{0.364} \approx 13.74 \text{ mm}\]

Step 5: Conclusion

Therefore, the focal length \( f \) for the 35-mm camera is approximately \( 49.45 \text{ mm} \) and for the digital camera is approximately \( 13.74 \text{ mm} \).

Key Concepts

Camera Focal Length

The focal length of a camera lens is a crucial concept in optics. It determines how a camera captures an image and how much of a scene is visible. A lens's focal length is the distance between the lens and the image sensor or film when the subject is in focus.
For a 35-mm camera, the `normal lens` focal length is the one that can capture the horizontal field of view the human eye can naturally see, about 40 degrees. This means that the camera lens must produce an image that fills up the entire width of the film, in this example, 36 mm.
The focal length (\(f\)) is calculated through trigonometric relationships by considering half the sensor's or film's width. The equation used is:

• \( \tan\left(\frac{\theta}{2}\right) = \frac{\text{half width of sensor}}{f} \)

Using this equation helps determine how far from the lens the sensor or film should be placed to create a clear image.

Trigonometric Functions

Trigonometric functions like tangent, sine, and cosine play vital roles in various calculations in geometry and physics. Specifically, in optics and this exercise, the tangent function helps understand the relationship between an object's visible angle and the dimensions captured by a lens.
To find the focal length, we use the tangent function because it relates the opposite side to the adjacent side in a right triangle. In this case, the `opposite` side is half the width of the camera’s sensor, and the `adjacent` side is the focal length.
Here's how you use it:

• Divide the total angle by two to find half of the field of view, converting it if needed, to radians.
• Apply the formula \( \tan\left(\frac{\theta}{2}\right) = \frac{\text{half width of sensor}}{f} \)
• Isolate \(f\) to calculate the focal length \(f = \frac{\text{half width of sensor}}{\tan\left(\frac{\theta}{2}\right)} \)

The tangent function is crucial here because it allows us to solve for physical dimensions using angular measurements.

Angle Conversion to Radians

Understanding angles is foundational for solving many geometrical problems, and conversions between degrees and radians are necessary for functions in calculus and trigonometry.
Degrees are most commonly used in everyday contexts, but radians provide a more natural mathematical measurement, especially for scientific calculations. One radian equals the angle created by taking the radius of a circle and wrapping it along the circle's edge, which is approximately 57.3 degrees.
To convert degrees to radians, use the conversion formula:

• \( \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \)

For example, a 40-degree field of view in degrees translates to radians using this equation:

• \(40^{\circ} \rightarrow \frac{\pi}{180} \times 40 = \frac{2\pi}{9} \text{ radians} \)

Radian conversion is essential when working with trigonometric functions since many formulas in calculus and physics expect angles in radians.