Problem 45 - University Physics with Modern Physics

Question

Figure 34.41 shows photographs of the same scene taken with the same camera with lenses of different focal length. If the object is 200 \(\mathrm{m}\) from the lens, what is the magnitude of the lateral magnification for a lens of focal length (a) \(28 \mathrm{mm} ;\) (b) 105 \(\mathrm{mm}\) ; (c) 300 \(\mathrm{mm}\) ?

Step-by-Step Solution

Answer

(a) 0.00014, (b) 0.000525, (c) 0.0015

Step 1: Introduction to Lateral Magnification Formula

To find the lateral magnification, we use the formula: \[m = \frac{-f}{ ext{object distance} - f}\]where \(m\) is the magnification, \(f\) is the focal length of the lens, and the object distance is given as 200 meters (200,000 mm). The negative sign indicates that the image is inverted.

Step 2: Calculate Magnification for 28 mm Focal Length

For a focal length of \(28\, \mathrm{mm}\), substitute \(f = 28\, \mathrm{mm}\) and the object distance \(= 200,000\, \mathrm{mm}\) into the formula:\[m = \frac{-28}{200000 - 28} \approx \frac{-28}{199972} \approx -0.00014\]Thus, the magnitude of the lateral magnification is approximately \(0.00014\).

Step 3: Calculate Magnification for 105 mm Focal Length

For a focal length of \(105\, \mathrm{mm}\), substitute \(f = 105\, \mathrm{mm}\):\[m = \frac{-105}{200000 - 105} \approx \frac{-105}{199895} \approx -0.000525\]The magnitude of the lateral magnification is approximately \(0.000525\).

Step 4: Calculate Magnification for 300 mm Focal Length

For a focal length of \(300\, \mathrm{mm}\), substitute \(f = 300\, \mathrm{mm}\):\[m = \frac{-300}{200000 - 300} \approx \frac{-300}{199700} \approx -0.0015\]Thus, the magnitude of the lateral magnification is approximately \(0.0015\).

Key Concepts

Focal Length

Focal length is a critical concept in understanding lenses and optical systems. It's defined as the distance between the lens and its focal point, where light rays either converge or appear to diverge. The focal length determines how strongly the lens converges or diverges light.
Short focal lengths have a greater bending effect, often resulting in a wider field of view, which is ideal for capturing larger scenes. On the other hand, long focal lengths offer a narrow field of view and are better suited for focusing on distant objects.
When determining the focal length's impact through lenses, remember:

• Shorter focal lengths capture more of the scene.
• Longer focal lengths make distant objects appear closer.

Understanding the focal length is essential for selecting the right lens for various photographic needs, like wide landscapes or close-up portraits.

Magnification Formula

The magnification formula is a powerful tool in optics that allows you to determine how much larger or smaller the image produced by a lens is compared to the actual object. The formula used for lateral magnification is:
\[ m = \frac{-f}{\text{object distance} - f} \]Here, \(m\) represents the magnification, \(f\) is the focal length, and the object distance is the distance from the object to the lens. The negative sign in the formula indicates that the image is inverted when the magnification is calculated.
Using this formula, you can easily calculate how the focal length affects the image size:

• If \(|m| < 1\), the image is smaller than the object.
• If \(|m| = 1\), the image is the same size as the object.
• If \(|m| > 1\), the image is larger than the object.

Understanding this formula can help you better grasp the relationship between the lens' focal length, the distance to the object, and the characteristics of the resulting image.

Optics

Optics is the branch of physics that deals with the behavior and properties of light. It includes the study of how lenses and mirrors form images.
In photography and in various optical devices like telescopes and microscopes, optics play a crucial role in image formation.
Key principles in optics include:

• Reflection: Light bouncing off a surface.
• Refraction: Light bending as it passes through different materials.
• Diffraction: Light spreading out after passing through small openings.

Understanding these principles helps in designing lenses that can produce clear and accurate images. Optics also encompasses how light interacts with lenses to create magnification, which is essential in understanding how images can appear larger or smaller.

Inverted Image

An inverted image is one that appears upside down compared to the original object. This phenomenon often occurs when using lenses, as they bend light in such a way that the top and bottom of the image switch places.
This can be well understood by considering the behavior of light rays as they pass through a lens. The light rays from the top of the object are bent downwards, while those from the bottom are bent upwards, creating an inverted image.
In photographic applications, many devices are built to correct this inversion, so the final image appears as expected on the viewfinder or screen. It's important to note:

• Lenses like those in cameras naturally invert images, yet mechanisms within the camera correct this.
• In simple optical setups, the inverted appearance is typical without further adjustment.

Recognizing why images invert is key to designing systems that use lenses, ensuring that the final image presentation meets the desired expectations.