Question

You are taking an eye exam. You stand 5 m from a board that has letters printed on it. The separation between two of the letters on the board is 1 cm. Assume that the light in the room has a wavelength of 600 nm.

a. If your pupil has a diameter of 7.5 mm, can you resolve the two letters, or do they blur together? (1 point)






b. What is the maximum distance at which you would be able to resolve the letters? (1 point)

Answer 1

Answer: a. To see if we can resolve the two letters, we need to calculate the angular resolution of the eye. Using the Rayleigh criterion, we have:

θ = 1.22 λ/D

where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the pupil. Substituting the given values, we get:

θ = 1.22 * (600 nm) / (7.5 mm) ≈ 0.01 radians

The angular separation between the two letters is:

φ = 1 cm / 5 m = 0.0002 radians

Since φ < θ, we can resolve the two letters and they won't blur together.

b. To find the maximum distance at which we can resolve the letters, we need to rearrange the Rayleigh criterion equation to solve for D:

D = 1.22 λ / θ

Substituting the given values, we get:

D = 1.22 * (600 nm) / 0.01 radians = 73.2 μm

The separation between the two letters is 1 cm or 10,000 μm, so the maximum distance at which we can resolve them is:

d = D * (distance to board) / (separation between letters) = 73.2 μm * (5 m) / (10,000 μm) = 0.037 m or 3.7 cm

Therefore, we can resolve the letters at distances up to 3.7 cm from the board.

Explanation: a. To see if we can resolve the two letters, we need to calculate the angular resolution of the eye. Using the Rayleigh criterion, we have:

θ = 1.22 λ/D

where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the pupil. Substituting the given values, we get:

θ = 1.22 * (600 nm) / (7.5 mm) ≈ 0.01 radians

The angular separation between the two letters is:

φ = 1 cm / 5 m = 0.0002 radians

Since φ < θ, we can resolve the two letters and they won't blur together.

b. To find the maximum distance at which we can resolve the letters, we need to rearrange the Rayleigh criterion equation to solve for D:

D = 1.22 λ / θ

Substituting the given values, we get:

D = 1.22 * (600 nm) / 0.01 radians = 73.2 μm

The separation between the two letters is 1 cm or 10,000 μm, so the maximum distance at which we can resolve them is:

d = D * (distance to board) / (separation between letters) = 73.2 μm * (5 m) / (10,000 μm) = 0.037 m or 3.7 cm

Therefore, we can resolve the letters at distances up to 3.7 cm from the board.