Problem 6 - College Physics

Question

. A camera with a 90 -mm-focal-length lens is focused on an object 1.30 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 0.0102 m closer to the film.

Step 1: Understand the Lens Formula

The lens formula is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \(f\) is the focal length, \(d_o\) is the object distance from the lens, and \(d_i\) is the image distance (distance from the lens to the film).

Step 2: Set Up Equations for Initial and Final Positions

Initially, the object distance \(d_{o1} = 1.3 \) m and the focal length \(f = 0.09\) m (convert 90 mm to meters). Using the lens formula, \( \frac{1}{0.09} = \frac{1}{1.3} + \frac{1}{d_{i1}} \). Similarly, for the final position with \(d_{o2} = 6.5\) m, use \( \frac{1}{0.09} = \frac{1}{6.5} + \frac{1}{d_{i2}} \).

Step 3: Solve for Initial Image Distance

Substitute \(f = 0.09\) m and \(d_{o1} = 1.3\) m into the lens formula: \( \frac{1}{0.09} = \frac{1}{1.3} + \frac{1}{d_{i1}} \). Calculate \(d_{i1}\). Solving gives \(d_{i1} \approx 0.1043\) m.

Step 4: Solve for Final Image Distance

Substitute \(d_{o2} = 6.5\) m into the lens formula: \( \frac{1}{0.09} = \frac{1}{6.5} + \frac{1}{d_{i2}} \). Calculate \(d_{i2}\). Solving gives \(d_{i2} \approx 0.0941\) m.

Step 5: Calculate the Change in Image Distance

Find the change in distance between the initial and final image positions: \( \Delta d_i = d_{i2} - d_{i1} = 0.0941 - 0.1043 \approx -0.0102\) m. This indicates the distance decreases.

Step 6: Determine Lens Movement Direction

Since \( \Delta d_i \) is negative, the lens must move closer to the film to refocus on the more distant object.

Key Concepts

Object Distance

The object distance, denoted as \(d_o\), refers to the distance between the object and the lens. It's one of the key components in understanding how lenses focus images. The lens formula \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) helps to establish the relationship between the object distance, image distance, and the focal length.When dealing with lenses in a camera, the object distance will change based on where the camera is pointed. For instance, in a photography scenario, if you're photographing a person 1.3 meters away, this is your object distance. If you then turn to photograph a mountain range that is 6.5 meters away, you've changed the object distance. Understanding how these changes affect focusing is crucial for clear photography.

• Initial object distance: 1.3 meters
• Final object distance: 6.5 meters

Changing the object distance requires adjusting the lens to maintain a sharp image.

Image Distance

The image distance \(d_i\) is the distance from the lens to the point where the image comes into focus. In cameras, this point is often where the film or a digital sensor is placed. When you adjust a camera's lens to bring different objects into focus, you're effectively changing the image distance. In practice, for an object that's closer to the lens, the image distance is longer compared to when the object is farther away. In our example, initially focusing on an object at 1.3 meters results in an image distance of approximately 0.1043 meters. When refocusing on an object that's 6.5 meters away, the image distance becomes approximately 0.0941 meters.

• Image Distance for 1.3 meters: \(d_{i1} \approx 0.1043\) meters
• Image Distance for 6.5 meters: \(d_{i2} \approx 0.0941\) meters

The change between these two distances is crucial for determining how the lens should be moved when changing focus.

Focal Length

Focal length, represented as \(f\), is an inherent property of a lens that indicates how strongly it converges or diverges light. It's the distance over which initially collimated (parallel) light is brought to a single focus point.In simpler terms, the focal length determines how zoomed in the photos will be. A shorter focal length allows for a wider angle of view, while a longer focal length zooms in on the subject. For the given exercise, our camera has a focal length of 90 mm, or 0.09 meters.Using the lens formula, this focal length aids in calculating the necessary adjustments to the lens when changing the focus from one object distance to another. The focal length remains constant during the focus adjustment, while the object and image distances change.In brief, the focal length is:

• Essential for determining the image focus
• A constant measurement when adjusting object and image distances
• 90 mm (0.09 m) in the case of our problem

Camera Focus Adjustment

Camera focus adjustment involves changing the distance between the lens and the sensor or film to produce a clear image. This adjustment compensates for different object distances measured from the lens. When you focus a camera on an object, you are adjusting this lens-to-sensor distance. This action is vital for clarity and sharpness in a photograph. In the example provided:

• To focus on a 1.3 m distant object, the image distance was 0.1043 m.
• To focus on a 6.5 m distant object, the image distance is reduced to 0.0941 m.
• The lens needs to move closer to the film or sensor by approximately 0.0102 meters to achieve sharp focus on the more distant object.

This change allows the camera to shift focus from near to far without losing image sharpness, ensuring that every photo captures clear details.