Problem 106 - University Physics with Modern Physics

Question

A camera with a 90 -mm-focal-length lens is focused on an object 1.30 \(\mathrm{m}\) from the lens. To refocus on an object 6.50 \(\mathrm{m}\) from the lens, by how much must the distance between the lens and the film be changed? To refocus on the more distant object, is the lens moved toward or away from the film?

Step-by-Step Solution

Answer

The lens moves 9.58 mm towards the film to refocus.

Step 1: Understand the Problem

We need to determine the change in lens-to-film distance when the camera lens refocuses from a near to a far object. The necessary information includes: the focal length (90 mm), the initial object distance (1.30 m), and the new object distance (6.50 m). We will use the lens formula for this purpose.

Step 2: Convert Units for Consistency

Convert the given object distances from meters to millimeters to match the focal length unit: - Initial object distance: 1.30 m = 1300 mm - New object distance: 6.50 m = 6500 mm.

Step 3: Apply the Lens Formula for Initial Object

The lens formula is: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \).- For the initial object, \( f = 90 \text{ mm} \), \( u = 1300 \text{ mm} \).- Rearrange to find the image distance \( v \): \[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{90} - \frac{1}{1300} \].- Calculate \( v \) using the above values to find the initial image distance.

Step 4: Calculate Initial Image Distance

Using the rearranged lens formula: \[ \frac{1}{v} = \frac{1300 - 90}{117000} \ {\Rightarrow} \ v \approx 100.86 \text{ mm} \] So, the image distance \( v \) for the initial object is approximately 100.86 mm.

Step 5: Apply the Lens Formula for New Object Distance

Now, repeat the process for the new object distance:- New object distance, \( u = 6500 \text{ mm} \).- Use the lens formula: \[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{90} - \frac{1}{6500} \].- Calculate the new image distance \( v \).

Step 6: Calculate New Image Distance

Calculate \( v \) using the rearranged formula: \[ \frac{1}{v} = \frac{6500 - 90}{585000} \ {\Rightarrow} \ v \approx 91.28 \text{ mm} \]So, the new image distance \( v \) for the new object is approximately 91.28 mm.

Step 7: Determine the Change in Image Distance

Find the difference between the two image distances:- Change in image distance \( = |100.86 \text{ mm} - 91.28 \text{ mm}| \).- Calculate the change: \( \ = 9.58 \text{ mm} \).

Step 8: Determine Direction of Lens Movement

Since the image distance decreased from 100.86 mm to 91.28 mm, the lens has moved closer to the film to refocus on the more distant object.

Key Concepts

Lens Formula

The lens formula is a fundamental equation in optics that relates the focal length of a lens to the distances of the object and the image it forms. This relationship is given by the formula: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \) where \( f \) is the focal length, \( v \) is the image distance (distance from the lens to the image), and \( u \) is the object distance (distance from the lens to the object).
This formula helps us determine how lenses focus light to form clear images. It is crucial in applications like photography, where precise focus adjustment is needed. Understanding this formula allows us to predict how changes in one parameter affect the others, helping us adjust lenses to capture clear images at different distances.

Focal Length

The focal length of a lens is a measure of how strongly it converges or diverges light. It is the distance between the lens and the point where parallel rays of light converge to a focus. In the context of the lens formula, the focal length \( f \) is a fixed value determined by the physical characteristics of the lens.
In a camera lens, the focal length can influence the field of view and magnification. A shorter focal length means a wider field of view, capturing more of the scene, while a longer focal length provides higher magnification. For our exercise, the lens had a focal length of 90 mm, which was consistent throughout the problem, influencing how we calculated the image distance for varying object distances.

Image Distance

The image distance, denoted by \( v \) in the lens formula, is the distance from the lens to the image formed. It is one of the variables in determining how an image focuses based on object distance and the lens's focal length.
Calculating the image distance is crucial in focusing a camera. For the initial object (1.3 m), the image distance was found to be approximately 100.86 mm. When refocusing the lens for an object at 6.5 m, the image distance became roughly 91.28 mm. Understanding how the image distance changes with varying object distances helps in adjusting the lens, ensuring the image captured is sharp and clear.

Object Distance

The object distance \( u \), another key element of the lens formula, represents how far the object is from the lens. Changes in the object distance can significantly alter the image distance due to the fixed nature of the focal length.
In the exercise, the object distance initially was 1.30 m (1300 mm) and later changed to 6.50 m (6500 mm). By applying the lens formula, we saw how this shift affected the image distance, prompting adjustments in the lens-to-film setup. A thorough grasp of how object distance impacts the overall optics system enables precise focusing by making the necessary physical adjustments to the lens position.