Problem 51 - Physics

Question

Late one night on a highway, a car speeds by you and fades into the distance. Under these conditions the pupils of your eyes have diameters of about \(7.0 \mathrm{~mm} .\) The taillights of this car are separated by a distance of \(1.2 \mathrm{~m}\) and emit red light (wavelength \(=660 \mathrm{nm}\) in vacuum). How far away from you is this car when its taillights appear to merge into a single spot of light because of the effects of diffraction?

Step-by-Step Solution

Answer

The car is approximately 10.4 kilometers away when its taillights appear to merge.

Step 1: Understanding the Problem

We are required to calculate the distance at which the car's taillights, due to diffraction effects, appear to merge into a single spot. This phenomenon occurs when the angular resolution limit is exceeded.

Step 2: Identify Applicable Formula

The angular resolution limit due to diffraction can be calculated using Rayleigh's criterion, given as: \[\theta = 1.22 \times \frac{\lambda}{d}\]where \(\lambda\) is the wavelength of light and \(d\) is the diameter of the eye's pupil.

Step 3: Convert Units Appropriately

Wavelength \(\lambda\) is given as \(660 \text{ nm} = 660 \times 10^{-9} \text{ m}\). The diameter of the pupil \(d\) is \(7 \text{ mm} = 7 \times 10^{-3} \text{ m}\).

Step 4: Substitute Values into Formula

Substitute the values into the formula to find \(\theta\):\[\theta = 1.22 \times \frac{660 \times 10^{-9} \text{ m}}{7 \times 10^{-3} \text{ m}}\]

Step 5: Calculate Angular Resolution \(\theta\)

After performing the calculation, we find:\[\theta \approx 1.15 \times 10^{-4} \, ext{radians}\]

Step 6: Contextualize Angular Resolution

The angular resolution \(\theta\) is the smallest angle at which two points can be distinctly observed. In our problem, it's the angle corresponding to the car's taillights looking like a single light.

Step 7: Use Geometry to Find Distance

Using the definition of the angle in relation to linear distance, the equation is given by:\[\theta = \frac{s}{L}\]where \(s = 1.2 \, \text{m}\) (distance between taillights) and \(L\) is the distance at which they merge. Rearrange to solve for \(L\):\[L = \frac{s}{\theta}\]

Step 8: Calculate Distance \(L\)

Substitute the known values to calculate \(L\):\[L = \frac{1.2 \text{ m}}{1.15 \times 10^{-4}}\approx 10434 \, \text{m}\]

Step 9: Final Step: Conclusion

Hence, the car is approximately \(10434\) meters, or \(10.4\) kilometers, away when the taillights appear as a single spot.

Key Concepts

Angular Resolution

Angular resolution essentially refers to the ability to distinguish two separate objects located at some distance away. It is an important concept in optics and is linked to how finely detail can be observed. Think about how a telescope allows us to view distant stars distinctly. The telescope's lens provides a certain level of angular resolution, which determines how finely those stars appear separately, rather than as a composite blur.

In the context of the given exercise, angular resolution is employed to figure out how far the taillights of a car can be separated but still appear merged together due to the diffraction of light. The smaller the angular resolution, the better the ability to differentiate between two distinct points, like the taillights of the distant car on the highway.

Rayleigh's Criterion

Rayleigh's Criterion is fundamental for understanding angular resolution. It provides the formula to calculate the minimum angle by which two close-point sources can be resolved as separate entities. This criterion is stated mathematically as: \[ \theta = 1.22 \frac{\lambda}{d} \]- \(\lambda\) is the wavelength of light - \(d\) represents the diameter of the optical component, in this case, the pupil of the eye.

The value '1.22' stems from detailed calculations involving the diffraction pattern produced by a circular aperture; this factor specifically applies to circular apertures such as the eye's pupil.

For night-time driving, this criterion explains why taillights at a certain distance appear as one light. As the distance between the two lights falls below this calculated angle, they can no longer be distinctly perceived.

Wavelength of Light

The wavelength of light is a crucial factor when considering diffraction and resolution capabilities. Light travels in waves, and the wavelength determines many of its properties. Visible light, like the red light emitted by the car's taillights, has a distinct wavelength that falls between 620 and 750 nanometers.

For the specific case in the exercise, the taillights are emitting red light at a wavelength of 660 nanometers. This is important because the wavelength influences how the light is diffracted through the pupil and, subsequently, impacts the angular resolution achieved by the eye.

Shorter wavelengths generally allow for better resolution because the angle \(\theta\) becomes smaller, meaning two objects can be distinguished at closer distances. It's one of the reasons why shorter-wavelength lights, like blue or violet, might be perceived as sharper.

Eye's Pupil Diameter

The diameter of the eye's pupil plays a significant role in how we perceive the world, especially under lower light conditions. Under these conditions, such as at night, the pupil dilates to allow more light to enter, which increases its diameter.

In the given problem, the pupil diameter is 7.0 mm. This larger opening at night allows for more light to reach the retina, helping in viewing distant objects.

According to Rayleigh's Criterion, as the diameter of the pupil increases, the angular resolution or the ability to distinguish close light sources improves. This is because a larger diameter results in a smaller angle \(\theta\), enhancing resolution. Hence, when a car speeds away at night, the resolution limit depends heavily on this pupil enlargement, impacting how the taillights are visually processed.