Problem 51 - University Physics with Modern Physics

Question

Photography. A 35 -mm camera has a standard lens with focal length 50 \(\mathrm{mm}\) and can focus on objects between 45 \(\mathrm{cm}\) and infinity. (a) Is the lens for such a camera a concave or a convex lens? (b) The camera is focused by rotating the lens, which moves it on the camera body and changes its distance from the film. In what range of distances between the lens and the film plane must the lens move to focus properly over the 45 \(\mathrm{cm}\) to infinity range?

Step-by-Step Solution

Answer

(a) The lens is a convex lens. (b) The lens moves between 50 mm and 56.25 mm from the film plane.

Step 1: Understand Lens Type

In cameras, the lens helps converge light to form a clear image. A 50 mm focal length in a 35mm camera is considered a standard lens. For cameras, the lenses need to converge light, pointing to a convex lens, as concave lenses diverge light.

Step 2: Use Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \), where \( f \) is the focal length, \( u \) is the object distance, and \( v \) is the image distance. In our case, \( f = 50 \) mm.

Step 3: Calculate Image Distance for Closest Focusing Distance

Set the object distance \( u = 450 \) mm (as 45 cm = 450 mm). Rearranging the lens formula, \( \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \), substitute \( f = 50 \) mm and \( u = 450 \) mm, then solve for \( v \).\[ \frac{1}{v} = \frac{1}{50} - \frac{1}{450} = \frac{450 - 50}{22500} = \frac{400}{22500} \] \( v = \frac{22500}{400} = 56.25 \) mm.

Step 4: Calculate Image Distance for Focusing at Infinity

When the camera is focused on infinity, \( u = \infty \). Thus, \( \frac{1}{u} = 0 \), reducing the lens formula to \( \frac{1}{v} = \frac{1}{f} \). Hence \( v = f = 50 \) mm.

Step 5: Determine Range of Lens Movement

Comparing the image distances calculated for closest (56.25 mm) and infinity (50 mm) focus, the lens must be capable of moving between these two positions. Therefore, the lens needs to move between 50 mm and 56.25 mm from the film plane.

Key Concepts

Convex Lens

In the world of photography, a convex lens plays a crucial role. A convex lens is a type of lens that is thicker at the center than at the edges. This shape causes light rays to converge at a point. For cameras, this is essential for forming sharp images on the film or sensor.
Convex lenses are the go-to choice for most photographic applications because they help gather and focus light from objects, creating clear images. Without this lens type, photographs would appear blurry.
In general, a convex lens is used in various optical devices like telescopes and glasses due to its light converging property.

• Convex lenses converge light rays to meet at a certain point.
• Essential for creating sharp camera images.
• Used in various optical devices.

Lens Formula

The lens formula is essential in photography and optics. It helps to relate the object distance, image distance, and focal length. This formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Where:
- \( f \) is the focal length,- \( u \) is the object distance,- \( v \) is the image distance.
Understanding this formula is important because it aids photographers in knowing how to adjust their camera settings to focus on different object distances. By rearranging this formula, photographers can calculate how the lens should be adjusted to focus correctly.
Whether you're aiming to capture distant landscapes or close-up portraits, the lens formula serves as the foundation for understanding how the lens behaves in different focusing situations.

Image Distance

The term 'image distance' refers to the distance between the lens and the image formed on the film or sensor. It's important because it determines where the camera lens should be positioned to achieve a clear image.
When using the lens formula, if you want to find the image distance \( v \), you rearrange the equation to get \( \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \).
For example, if your camera is focusing on an object 45 cm away, you plug the numbers into the formula to find the image distance. The calculation reveals that the lens must be at a specific position from the film to ensure clarity.
This understanding allows photographers to adjust the lens movement between the closest focus and infinity focus, as seen in cameras with zoom or focus adjustments.

• Image distance helps determine lens positioning.
• Critical for achieving clear, focused images.
• Connects to both focal length and object distance in lens formula.

Focal Length

Focal length is a key characteristic of a lens and defines the field of view of a camera. It's the distance from the center of the lens to the focal point, where light rays converge to form a sharp image.
For example, a 50 mm lens is often standard in many cameras as it offers a natural perspective similar to human vision. A shorter focal length means a wider field of view, while a longer one results in a narrow and more zoomed-in view.
Photographers need to understand focal length to select the right lens for their subject matter. It influences both the composition of the photograph and the detail captured.
The focal length also plays a crucial role in the lens formula, influencing both the object and image distances. Understanding how focal length affects image capture is essential for anyone looking to effectively use a camera.

• Focal length impacts field of view and perspective.
• Important for lens selection based on subject.
• Integral part of the lens formula.