Problem 28 - Physics

Question

A hunter who is a bit of a braggart claims that, from a distance of \(1.6 \mathrm{~km}\), he can selectively shoot either of two squirrels who are sitting ten centimeters apart on the same branch of a tree. What's more, he claims that he can do this without the aid of a telescopic sight on his rifle. (a) Determine the diameter of the pupils of his eyes that would be required for him to be able to resolve the squirrels as separate objects. In this calculation use a wavelength of \(498 \mathrm{~nm}\) (in vacuum) for the light. (b) State whether his claim is reasonable, and provide a reason for your answer. In evaluating his claim, consider that the human eye automatically adjusts the diameter of its pupil over a typical range of 2 to \(8 \mathrm{~mm}\), the larger values coming into play as the lighting becomes darker. Note also that under dark conditions, the eye is most sensitive to a wavelength of \(498 \mathrm{~nm}\).

Step-by-Step Solution

Answer

The hunter's claim is unreasonable as the required pupil diameter is 9.7 cm, much larger than human capability.

Step 1: Understanding Rayleigh Criterion

To determine if the hunter can resolve the squirrels independently with the naked eye, we use the Rayleigh criterion. This criterion states that the minimum resolvable angle \( \theta \) is given by the formula \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of light (\(498 \mathrm{~nm} = 498 \times 10^{-9} \mathrm{~m}\)) and \( D \) is the diameter of the pupil.

Step 2: Calculate the Angular Separation

The angular separation \( \theta_{s} \) of the squirrels is determined by the formula \( \theta_{s} = \frac{d}{L} \), where \( d = 0.1 \mathrm{~m} \) is the distance between the squirrels, and \( L = 1600 \mathrm{~m} \) is the distance to the squirrels. Hence, \( \theta_{s} = \frac{0.1}{1600} \).

Step 3: Equating Rayleigh Criterion and Angular Separation

For the hunter to resolve the squirrels separately, the Rayleigh criterion angle \( \theta \) must at least match the angular separation \( \theta_{s} \). Set \( 1.22 \frac{\lambda}{D} = \frac{0.1}{1600} \). Substitute \( \lambda = 498 \times 10^{-9} \mathrm{~m} \) to solve for \( D \): \( D \geq 1.22 \times \frac{498 \times 10^{-9}}{0.1/1600} \).

Step 4: Solve for Pupil Diameter

Solving the equation from the previous step gives \( D \geq 1.22 \times \frac{498 \times 10^{-9} \times 1600}{0.1} = 1.22 \times 7.968 \times 10^{-6} \approx 9.7 \mathrm{~cm} \). So the eye would need a pupil diameter of at least \( 9.7 \mathrm{~cm} \) to resolve the squirrels without aid.

Step 5: Analyzing the Claim

The calculated pupil diameter \( (9.7 \mathrm{~cm}) \) exceeds the physical capability of the human eye, which ranges from \( 2 \) to \( 8 \mathrm{~mm} \). Thus, the hunter's claim is unreasonable without the aid of additional equipment like a telescope.

Key Concepts

Rayleigh Criterion

The Rayleigh Criterion is a principle used to determine the ability of an optical system, like the human eye, to distinguish between two closely spaced objects. According to this criterion, two points are resolvable when the principal diffraction maximum of one image coincides with the first minimum of the other image. In simpler terms, for two objects to be seen as separate, their images must be sufficiently distinct.
In mathematical terms, the minimum resolvable angle \( \theta \) is given by \( \theta = 1.22 \frac{\lambda}{D} \), where:

• \( \lambda \) is the wavelength of light — in this case, 498 nm.
• \( D \) is the diameter of the optical aperture — here, the hunter's pupil.

For the calculation of the hunter’s ability to distinguish between the two squirrels without optical aid, the Rayleigh Criterion helps establish the smallest angular separation his eyes can resolve.

Angular Separation

Angular Separation refers to the apparent distance between two objects as seen from a given vantage point, measured as an angle. This concept is essential when discussing resolution because it tells us how two separate objects appear from a certain distance.
For the hunter, the separation between the squirrels provides a practical example of angular separation. Calculating this involves the formula:
\[ \theta_{s} = \frac{d}{L} \]
where:

• \( d \) represents the physical distance between the squirrels (0.1 meters here)
• \( L \) is the observer's distance from the objects (1600 meters in this problem)

Such an understanding allows us to know if the angular separation is within the resolving power determined by the Rayleigh Criterion.

Human Eye Limits

The human eye has physical limitations in terms of both pupil size and resolution power. Typically, the human pupil's diameter ranges from 2 mm to 8 mm depending on lighting conditions. The larger sizes occur in low light conditions, aiding resolution to some extent.
In the scenario with the hunter, aiming to resolve squirrels 10 cm apart at 1.6 km without a telescopic sight, the eye would need a pupil diameter calculated at approximately 9.7 cm based on the Rayleigh Criterion and angular separation considerations. This value greatly exceeds the actual physiological limits of the human pupil.
Consequently, the hunter's claim to resolve the squirrels unassisted is physically impossible under known human eye constraints, underscoring the need for optical aids such as telescopes or binoculars for such tasks.