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

[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

[tex]\theta[/tex]min=1.22[tex]\lambda[/tex]/D
where [tex]\lambda[/tex] 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 [tex]\theta[/tex]2. found that the distance from the eye to the car would be 60tan[tex]\theta[/tex]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?

 

[Mr.Amin]

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.)

 

[nlightner]

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.