Problem 53 - Physics

Question

The pupil of an eagle's eye has a diameter of \(6.0 \mathrm{~mm}\). Two field mice are separated by \(0.010 \mathrm{~m}\). From a distance of \(176 \mathrm{~m}\), the eagle sees them as one unresolved object and dives toward them at a speed of \(17 \mathrm{~m} / \mathrm{s}\). Assume that the eagle's eye detects light that has a wavelength of \(550 \mathrm{~nm}\) in vacuum. How much time passes until the eagle sees the mice as separate objects?

Step-by-Step Solution

Answer

The eagle will take approximately 2.2 seconds to see the mice as separate objects.

Step 1: Understanding Rayleigh Criterion

The limit of resolution for optical instruments, such as an eagle's eye, can be determined using the Rayleigh criterion formula given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \theta \) is the minimum angular separation the eye can resolve, \( \lambda \) is the wavelength of light (\(550 \times 10^{-9} \mathrm{~m}\)), and \( D \) is the diameter of the aperture (the eagle's pupil, \(6.0 \times 10^{-3} \mathrm{~m}\)).

Step 2: Calculate Angular Resolution

Using the Rayleigh criterion: \[ \theta = 1.22 \frac{550 \times 10^{-9}}{6.0 \times 10^{-3}} \] Calculate \( \theta \) to find the minimum angular separation that the eagle can resolve.

Step 3: Convert Angular Resolution to Distance

Using the formula \( \theta = \frac{s}{d} \), where \( s \) is the separation distance between the mice (\(0.010 \mathrm{~m}\)) and \( d \) is the distance from the eagle to the mice, solve for \( d \): \( d = \frac{s}{\theta} \).

Step 4: Find Distance Reduction Required

Given that the initial distance from the eagle to the mice is \(176 \mathrm{~m}\), calculate the decrease in distance required for the eagle to resolve the mice: \( \Delta d = 176 \mathrm{~m} - d \).

Step 5: Calculate Time to Reach Desired Distance

The speed of the eagle is given as \(17 \mathrm{~m/s}\). Using the formula \( t = \frac{\Delta d}{v} \), where \( \Delta d \) is the required distance reduction and \( v \) is the speed of the eagle, calculate the time \( t \).

Key Concepts

Angular Resolution

Angular resolution is a vital concept in understanding how we and other creatures perceive detail from a distance. Defined by the angle between two lines of sight to two distinct objects, it determines how well a visual system can differentiate between two closely spaced objects. Angular resolution depends on the optics of the observing tool, be it natural like an eagle's eye or a human-made instrument like a telescope. In optics, the smaller the angle, the better the resolution, which means the observer can distinguish finer details. When calculated using the Rayleigh criterion, we identify the smallest angle of separation that can be resolved by the optical system. For an eagle, which has a smaller pupil diameter compared to telescopes, the angular resolution is limited. Yet, its keen eyesight far surpasses that of humans, thanks to its evolutionary adaptations for hunting.

Optical Instruments

Optical instruments are devices engineered to extend the power of natural vision. They include microscopes, telescopes, and even the eyes of living creatures. Each of these instruments operates under the principles of optics, utilizing lenses or mirrors to manipulate light. The eagle's eye, an example of an advanced natural optical device, comprises features that enhance its resolution capacity. These include a high density of visual receptors within the retina and a large lens capable of adjusting focus rapidly as the eagle maneuvers mid-flight. Through understanding optical instruments, scientists improve technology, creating devices that mimic these extraordinary natural systems. By learning from organisms like the eagle, engineers enhance equipment from simple cameras to complex astronomical telescopes.

Wavelength of Light

The wavelength of light is a critical factor influencing the resolution of optical instruments. Light travels in waves, and its wavelength is the distance between consecutive peaks of these waves. It can range from shorter lengths, like ultraviolet, to longer, like infrared. In the visible spectrum, wavelengths range approximately from 400 to 700 nanometers (nm). The problem we're tackling uses a wavelength of 550 nm, which corresponds to green light. The choice of wavelength affects the Rayleigh criterion. Shorter wavelengths can improve resolution, allowing the optical system to differentiate between points placed closely together. This is why researchers carefully select particular wavelengths to optimize resolution in microscopes and other optical devices.

Limit of Resolution

The limit of resolution, sometimes called the resolving power, is the point at which an optical system can just resolve two points as distinct entities. It's a central aspect of the Rayleigh criterion, which helps determine whether an instrument can distinguish between two separate objects at a certain distance. For an eagle, this limit denotes how closely two prey items, such as field mice, can be while still appearing as separate objects, even from a great height. The eagle's ability to resolve such fine detail is crucial for successful predation. The limit of resolution is influenced by several factors, including the diameter of the eye's pupil, the wavelength of light entering the eye, and the distance between the observer and the objects. These aspects interplay to define the optical limits that both natural and artificial systems can achieve.