Problem 39 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A nature photographer wishes to photograph a \(38-\mathrm{m}\) tall tree from a distance of \(65 \mathrm{~m}\). What focal-length lens should be used if the image is to fill the 24 -mm height of the sensor?

Step-by-Step Solution

Answer

The focal length of the lens should be approximately 41.05 mm.

Step 1: Understand the Problem

The photographer wants an image of a 38-meter tall tree to fill a 24-mm sensor. We need to determine which focal length will result in the image of the tree fitting exactly within the sensor size.

Step 2: Use the lens formula

We use the concept of similar triangles and the lens formula to determine the focal length. The ratio of the height of the image to the height of the object is equal to the ratio of the focal length to the object distance.\[\frac{h_i}{h_o} = \frac{f}{d_o}\]Where:- \(h_i\) = height of the image (sensor height) = 24 mm- \(h_o\) = height of the object (tree) = 38,000 mm- \(d_o\) = distance from the object to the lens = 65,000 mm- \(f\) = focal length in mm

Step 3: Solve for the Focal Length

Rearrange the equation to solve for \(f\):\[f = \frac{h_i \times d_o}{h_o}\]Substitute the known values into the equation:\[f = \frac{24 \times 65,000}{38,000} \text{ mm}\]Calculate \(f\):\[f = \frac{1,560,000}{38,000} \text{ mm} \approx 41.05 \text{ mm}\]

Step 4: Conclusion

The photographer should use a lens with a focal length of approximately 41.05 mm to ensure the image of the 38-meter tall tree exactly fits the 24-mm sensor height.

Key Concepts

Focal Length

The focal length of a lens is a fundamental concept in optics. It represents the distance from the lens to its focal point, where parallel rays of light converge.
This distance determines how the lens bends the light and affects the size and clarity of the image. In simpler terms, the focal length helps us understand how much of a scene will be captured and how large it will appear on a given sensor.

• A shorter focal length (like 18 mm) captures a wider view and is typical for landscape photography.
• Conversely, a longer focal length (like 200 mm) magnifies the subject, which is useful for capturing distant objects in detail.

For our exercise, calculating the right focal length ensures that the tree fills the 24-mm sensor just right.

Lens Formula

The lens formula is a mathematical equation that relates the object distance (\(d_o\)), image distance, and the focal length (\(f\)) of a lens. For a clear image formation, the lens formula is generally expressed as:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i},\]where,

• \(f\) is the focal length,
• \(d_o\) is the distance between the object and the lens,
• \(d_i\) is the distance from the lens to the image.

In our specific case, the lens formula is adjusted to relate the size of the image to the size of the object, incorporating these distances as well. This involves interpreting the proportions that allow using \(\frac{h_i}{h_o} = \frac{f}{d_o}\), making the focal length derivable in terms of height and object distance.

Similar Triangles

Similar triangles are crucial in deriving relationships between different measurements in optics. When applied to lenses and image formation, the principle of similar triangles helps establish the relationship between the real object and its captured image.
In the context of the exercise, similar triangles allow us to use the object (tree's height) and the image (sensor's height) to figure out the unknown, which is the focal length.
By setting up the proportion \(\frac{h_i}{h_o} = \frac{f}{d_o}\), we state that the ratios of heights are equivalent to the ratios of distances. This is due to the angles of light rays remaining consistent between object and image formation, which forms geometrically similar triangles.
This understanding helps simplify many optical problems, converting physical dimensions into manageable mathematical expressions.