Problem 111 - University Physics with Modern Physics

Question

To determine whether a frog can judge distance by means of the amount its lens must move to focus on an object, researchers covered one eye with an opaque material. An insect was placed in front of the frog, and the distance that the frog snapped its tongue out to catch the insect was measured with high-speed video. The experiment was repeated with a contact lens over the eye to determine whether the frog could correctly judge the distance under these conditions. If such an experiment is performed twice, once with a lens of power -9-D and once with a lens of power -15-D, in which case does the frog have to focus at a shorter distance, and why? (a) With the -9-D lens; because the lenses are diverging, the lens with the longer focal length creates an image that is closer to the frog. (b) With the -15-D lens; because the lenses are diverging, the lens with the shorter focal length creates an image that is closer to the frog. (c) With the -9-D lens; because the lenses are converging, the lens with the longer focal length creates a larger real image. (d) With the -15-D lens; because the lenses are converging, the lens with the shorter focal length creates a larger real image.

Step-by-Step Solution

Answer

(b) With the -15-D lens; shorter focal length means image is closer.

Step 1: Identifying Lens Types

The problem involves lenses with negative powers (-9-D and -15-D), which are diverging or concave lenses. These lenses create virtual images that are located closer to the lens when the focal length (absolute value of power) is lower.

Step 2: Understanding Lens Power and Focal Length Relationship

The power of a lens (P) is given by the formula \( P = \frac{1}{f} \), where \( f \) is the focal length in meters. For a diverging lens, the focal length is negative. Therefore, a lens with a power of -15-D has a shorter focal length than one with -9-D.

Step 3: Evaluating the Impact of Focal Length on Virtual Image Distance

A shorter focal length (higher absolute power, e.g., -15-D) in a diverging lens means the virtual image formed is closer to the lens. Thus, the frog would need to focus at a shorter distance when using the -15-D lens compared to the -9-D lens.

Step 4: Conclusion Based on Lens Formula

Considering the lens formula and the properties of diverging lenses, option (b) is correct: with the -15-D lens, the frog has to focus at a shorter distance because the lens with the shorter focal length creates an image closer to the frog than the lens with a longer focal length.

Key Concepts

Diverging Lens

A diverging lens, also known as a concave lens, spreads light rays apart. Unlike converging lenses that focus light to a point, diverging lenses make parallel light rays spread out as if they came from a point behind the lens. This results in a virtual image.

• These lenses are typically thinner at the center than at the edges.
• They are characterized by having a negative focal length.
• Diverging lenses are commonly used to correct nearsightedness.

Understanding how a diverging lens works can help explain their behavior in optical applications, such as focusing images closer to or further from the lens.

Lens Power

Lens power is a crucial concept in optics, indicating how strongly a lens can bend light. The power of a lens is measured in diopters (D) and is calculated using the formula:
\[ P = \frac{1}{f} \]where \( P \) is the power in diopters and \( f \) is the focal length in meters.

• A negative power indicates a diverging lens.
• Higher absolute values of lens power mean a stronger lens.
• For instance, a -15-D lens has more power than a -9-D lens.

Lens power is essential for determining how much a lens will affect the path of light passing through it, influencing the formation and position of images.

Focal Length

The focal length of a lens is the distance from the lens to the point where light rays converge or appear to diverge. In diverging lenses, this point is virtual and located on the same side of the lens as the incoming light.

• It is negative for diverging lenses.
• Shorter focal lengths indicate stronger lenses, providing bigger divergence.
• In our example, a focal length associated with -15-D is shorter than -9-D.

This concept helps determine how lenses manipulate light, as shorter focal lengths result in virtual images forming closer to the lens itself.

Virtual Image

A virtual image is an image that appears to be at a location from which light does not actually come. Diverging lenses always form virtual images since they diverge light rays. These images can't be projected onto a screen because the light rays do not physically meet.

• Virtual images are upright compared to the object.
• Formed on the same side as the object in diverging lenses.
• The size of the virtual image can vary based on the lens's focal length.

Virtual images are crucial for understanding how the brain perceives light and dimensions, enhancing our interpretation of distances in objects viewed through lenses.