# At what distance can your eye no longer resolve two headlights?

• limonysal
In summary, the question is asking at what distance the two headlights of an oncoming car will be marginally resolved by the lens of the eye, assuming a wavelength of 600 nm and an index of refraction of 1.33. Using the equation for angular resolution and a diameter of 7 mm for the pupil, the calculated distance is 6.574*10^7 cm. However, this is more than what the eye can resolve, possibly due to the lens being diffraction limited.
limonysal

## Homework Statement

Once dark adapted, the pupil of your eye is approximately 7 mm in diameter. The headlights of an oncoming car are 120 cm apart. If the lens of your eye is diffraction limited, at what distance are the two headlights marginally resolved? Assume a wavelength of 600 nm and that the index of refraction inside the eye is 1.33.

## Homework Equations

$$\theta$$min=1.22$$\lambda$$/D
where $$\lambda$$ is the wavelength, D is the diameter of the lens, and theta min is the angular resolution of a lens.

## The Attempt at a Solution

So I've tried finding the distance according to the angle...but I'm confused. Why is the index of refraction included?

I got 1.0457*10^-4 as the angle. Then divided that by half, 5.229*10^-5, call it $$\theta$$2. found that the distance from the eye to the car would be 60tan$$\theta$$2=6.574*10^7 cm. The question also said that the answer you would get would be more than what the eye can resolve, but I still think this might be off. Maybe it has to do with the lens being diffraction limited?

Your eye not diffraction limited of course.
Usually problems do not hand you extra information in first/second year physics.
What happens to light when it enters a lens?
(Hint: The equation you are using works well for mirrors telescopes.)

I would like to clarify a few things about the question and your attempt at a solution. Firstly, the index of refraction is included because it affects the speed of light inside the eye, which in turn affects the angular resolution of the lens. Therefore, it is an important factor to consider in this scenario.

Secondly, your calculation for the angular resolution using the equation \thetamin=1.22\lambda/D is correct. However, the value you obtained for the angle is in radians and needs to be converted to degrees. This can be done by multiplying the value by 180/pi. This gives an angular resolution of 0.003 degrees.

Next, to find the distance at which the two headlights are marginally resolved, we can use the formula d = 2Dtan\theta2, where d is the distance between the two headlights and D is the distance from the eye to the car. Substituting the values, we get d = 120 cm and \theta2 = 0.003 degrees. This gives a distance of approximately 4 km.

Lastly, the question mentions that the answer will be more than what the eye can resolve. This is because the human eye has a maximum angular resolution of about 0.02 degrees. Therefore, even though the calculated distance is within the range of what the eye can physically see, the two headlights will still appear as one blurred object to the eye.

In conclusion, the distance at which the two headlights are marginally resolved is approximately 4 km, taking into consideration the diffraction-limited lens and the index of refraction inside the eye. However, due to the limitations of the human eye, the two headlights will still appear as one blurred object at this distance.

## 1. What is the maximum distance at which the human eye can resolve two headlights?

The maximum distance at which the human eye can resolve two headlights depends on various factors such as the size and brightness of the headlights, the lighting conditions, and the individual's visual acuity. On average, it is believed that the human eye can resolve two headlights at a distance of approximately 1 mile (1.6 kilometers).

## 2. How does the size of the headlights affect the distance at which they can be resolved?

The larger the headlights, the further the human eye can resolve them. This is because larger objects appear larger in the visual field, making it easier for the eye to distinguish between them. However, the size of the headlights alone is not the only determining factor.

## 3. Does the brightness of the headlights make a difference in their resolution distance?

Yes, the brightness of the headlights does play a role in the distance at which they can be resolved. The brighter the headlights, the further they can be resolved by the human eye. This is because brighter objects create a stronger contrast against their surroundings, making them easier to distinguish.

## 4. Can the distance at which the human eye can resolve two headlights be improved?

Yes, there are certain measures that can be taken to improve the distance at which the human eye can resolve two headlights. These include using brighter headlights, ensuring proper lighting conditions, and correcting any visual impairments that may affect visual acuity.

## 5. Is there a limit to how far the human eye can resolve two headlights?

Yes, there is a limit to how far the human eye can resolve two headlights. This is due to the limited capabilities of the human eye and the fact that light waves eventually disperse and become too faint to be detected by the eye. However, with advancements in technology, there are now devices such as binoculars and telescopes that can extend the resolution distance beyond what the human eye is capable of.

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