Problem 83 - Physics for Scientists and Engineers with Modern Physics

Question

A \(200-\mathrm{mm}\) -focal-length lens can be adjusted so that it is \(200.0 \mathrm{~mm}\) to \(206.4 \mathrm{~mm}\) from the film. For what range of object distances can it be adjusted?

Step-by-Step Solution

Answer

Object distances range from infinity to approximately \(6885\,\text{mm}\).

Step 1: Understand the Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. For a lens with a focal length of \( 200\,\text{mm} \), we can plug in the values to find the object distance range.

Step 2: Calculate Object Distance for Minimum Image Distance

Let's start with the minimum image distance of \( 200.0\,\text{mm} \). Using the lens formula:\[ \frac{1}{200} = \frac{1}{d_o} + \frac{1}{200} \]Simplify to find \( d_o \) by solving \( \frac{1}{d_o} = 0 \), which gives an infinite object distance. This means the lens is focused at infinity.

Step 3: Calculate Object Distance for Maximum Image Distance

For the maximum image distance of \( 206.4\,\text{mm} \):\[ \frac{1}{200} = \frac{1}{d_o} + \frac{1}{206.4} \]Rearrange to find \( d_o \):\[ \frac{1}{d_o} = \frac{1}{200} - \frac{1}{206.4} \]Calculate \( \frac{1}{d_o} \), and then take the reciprocal to get \( d_o \approx 6885\,\text{mm} \). This is the nearest object distance the lens can focus on at this image distance.

Key Concepts

Focal Length

Focal length is a fundamental concept in optics, determining how strongly a lens converges or diverges light. Simply put, the focal length is the distance from the center of a lens to its focal point, where parallel rays of light meet. In this exercise, we deal with a lens with a focal length of 200 mm.

• The focal length dictates how much space between the lens and the film or sensor you need for clear focus.
• For lenses with longer focal lengths, objects appear larger but cover less area in the frame.
• A 200 mm focal length is typical for a telephoto lens, excellent for focusing on distant subjects.

The focal length determines both magnification and the field of view. It directly ties into the lens formula used to calculate object and image distances.

Object Distance

Object distance represents how far an object is from the lens. In photography or optics, knowing where your subject is relevant is crucial for a good focus. The lens formula connects it directly with image distance and focal length:
\(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)

• In this exercise, we found the object distance for two scenarios using the image distances.
• When calculating the minimum image distance being 200 mm, the result showed that the object can be infinitely far, meaning the lens is set to focus on infinity.
• For the maximum image distance of 206.4 mm, the object distance is computed to be around 6885 mm, which means this is the closest a subject can be for the lens to focus.

Understanding object distance helps in setting lenses correctly to capture sharp images of subjects whether near or far.

Image Distance

Image distance is another key element in understanding the behavior of lenses. It indicates how far the captured image is placed from the lens, typically onto a camera sensor or film. Variations in image distance affect how objects are focused through the lens formula.

• Smaller image distances often require greater object distances to maintain clarity and focus.
• In the given exercise, changing the image distance between 200 mm and 206.4 mm allowed for different object distances; from focusing on infinity to about 6885 mm.
• The closer the image distance to the focal length, the more the setup tends towards focusing on very distant objects.

Mastering the balance between image distance and the other parameters ensures clear and focused images under various circumstances.