Problem 83 - Physics for Scientists and Engineers with Modern Physics

Question

A 200 -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

The object distance range is 6450 mm to infinity.

Step 1: Understanding the Lens Formula

The lens formula is given by the equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the lens, \( d_o \) is the object distance, and \( d_i \) is the image distance from the lens to the film. In this problem, \( f = 200 \, \mathrm{mm} \), and \( d_i \) varies from 200.0 mm to 206.4 mm.

Step 2: Setting up Equations for Object Distances

We need to find the range of object distances \( d_o \) for two extreme positions of the image distance \( d_i \). First, we set up the equation for \( d_i = 200.0 \, \mathrm{mm} \):\[\frac{1}{200} = \frac{1}{d_o} + \frac{1}{200}\]. Simplifying, we get \( \frac{1}{d_o} = 0 \), meaning \( d_o \) tends towards infinity. Next, for \( d_i = 206.4 \, \mathrm{mm} \) we set up the equation:\[\frac{1}{200} = \frac{1}{d_o} + \frac{1}{206.4}\].

Step 3: Calculating Object Distance for Maximum Image Distance

For \( d_i = 206.4 \, \mathrm{mm} \), solve \( \frac{1}{200} = \frac{1}{d_o} + \frac{1}{206.4} \):Subtract \( \frac{1}{206.4} \) from \( \frac{1}{200} \):\[\frac{1}{d_o} = \frac{1}{200} - \frac{1}{206.4} = \frac{206.4 - 200}{200 \times 206.4} = \frac{6.4}{200 \times 206.4}\].Hence, \( d_o = \frac{200 \times 206.4}{6.4} \).

Step 4: Finding Numerical Value for Object Distance

Calculate \( d_o \) using the expression:\[ d_o = \frac{200 \times 206.4}{6.4} \equiv 6450 \, \mathrm{mm}. \]

Step 5: Interpreting the Result

Our calculations show that the minimum object distance is approximately 6450 mm. As \( d_o \) for \( d_i = 200 \) mm tends towards infinity, the range of object distances is from around 6450 mm to infinity.

Key Concepts

Focal Length

The focal length is a crucial property of a lens that determines where light rays converge to form a clear image. In simple terms, it is the distance between the lens and its focus, where parallel light beams meet after passing through the lens. For the given lens, the focal length is specified as 200 mm, meaning it will focus light rays 200 mm away from the lens. This measurement plays a central role in calculations involving the lens formula.

• Whenever you know the focal length, you can predict how a lens interacts with light.
• A shorter focal length focuses light at a closer point, often producing a wider field of view.
• A longer focal length does the opposite, providing a narrower field of view but greater magnification.

This understanding helps in setting up the lens formula to determine how far away an object should be placed to get a sharp image at varying image distances.

Object Distance

Object distance (\(d_o\)) is the distance between the object you want to focus on and the lens. Understanding how to measure and calculate this distance is vital in optics, as it affects how clear and accurate your image results will be. To find the object distances for varying image distances, the lens formula can be used:equation: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This equation requires rearranging based on what you need to solve. When the image distance (\(d_i\)) is very close to the focal length (like the case for 200 mm), you may find that \(d_o\) tends towards infinity.

• This implicates that there's a nearly infinite distance at which objects can still project clear images on the film.
• At a maximum image distance of 206.4 mm, using the above formula and solving, you find the minimum \(d_o\) is 6450 mm.
• The range of object distances thus stretches from 6450 mm to infinity.

The flexibility of object distances is dependent on lens adjustments, making this calculation vital for photographers and scientists alike.

Image Distance

Image distance (\(d_i\)) tells us how far the image forms from the lens. It's a vital part of understanding how lenses work to create the images we see, whether through a camera or a projector. In the problem at hand, \(d_i\) varies between 200 mm and 206.4 mm. These values are plugged into the lens formula to calculate possible object distances.

• When \(d_i = 200\) mm, any object distance practically yields an image because it matches the lens's focal length ideally.
• Adjustments to the lens can allow for slight variations, like extending to \(d_i = 206.4\) mm.

Knowing \(d_i\) is essential for adjusting lenses to capture sharp images, whether focusing on distant stars or close-up details. Proper manipulation of \(d_i\) helps optimize lenses for diverse use cases, adjusting either for more depth of field or precision focusing.