Suerat made paintings in which separate dots of color are placed close together on the canvas, and from a distance they merge in the eye to form an image. Consider such a painting with the dots in the painting separated by 1.7 mm and assume that the wavelength of the light is ëvacuum = 500 nm. Find the distance at which the dots can just be resolved by each of the following.

(a) the eye (pupils diameter 2.5 mm)

(b) a camera (aperture diameter 25 mm)

Use theta = 1.22 (wavelength)/Diameter for the resolution limit in radians. This is referred to as the Dawes' (or Airy) resolution limit.

Let x = 1.7^10^-3 m be the spot size and R be the distance from the painting.

Require that
1.22 (wavelength)/D = x/R

and then solve for R.
wavelength = 500*10^-9 m

(a) When D = 2.5*10^-3 m.
(1.22)*500*10^-9/2.5*10^-3 = 1.7*10^-3/R

Solve for R.

I get R = 7 meters

(b) Repeat the calculation with the larger value of aperture diameter, D.

To find the distance at which the dots can just be resolved by the eye and the camera, we can use a concept called the Rayleigh criterion. According to the Rayleigh criterion, two objects can just be resolved if the central maximum of one object's diffraction pattern coincides with the first minimum of the other object's diffraction pattern.

Using the Rayleigh criterion, we can determine the distance at which the dots can just be resolved by each device.

(a) For the eye with a pupil diameter of 2.5 mm:
The angular resolution of the eye (θ) can be calculated using the equation:
θ = 1.22 * λ / D
where λ is the wavelength of light and D is the diameter of the pupil.

Substituting the values:
λ = 500 nm = 500 * 10^-9 m
D = 2.5 mm = 2.5 * 10^-3 m

θ = 1.22 * (500 * 10^-9) / (2.5 * 10^-3)
= 1.22 * 0.0005 / 0.0025
= 0.244 radians

Now, to find the distance at which the dots can just be resolved, we can calculate the separation (S) between the dots using the equation:
S = r * θ
where r is the distance from the eye to the painting.

In this case, we want to find the maximum distance at which the dots can just be resolved, so we use the minimum separation between the dots (1.7 mm) as our reference (S = 1.7 mm).

1.7 mm = r * 0.244
r = 1.7 mm / 0.244
r ≈ 6.97 mm

Therefore, the maximum distance at which the dots can just be resolved by the eye is approximately 6.97 mm.

(b) For the camera with an aperture diameter of 25 mm:
Similar to the calculation for the eye, we can use the same formula to calculate the angular resolution (θ) for the camera:
θ = 1.22 * λ / D

Substituting the values:
λ = 500 nm = 500 * 10^-9 m
D = 25 mm = 25 * 10^-3 m

θ = 1.22 * (500 * 10^-9) / (25 * 10^-3)
= 1.22 * 0.0005 / 0.025
= 0.0244 radians

Again, using the minimum separation between the dots (1.7 mm) as our reference, we can calculate the maximum distance at which the dots can just be resolved:
1.7 mm = r * 0.0244
r = 1.7 mm / 0.0244
r ≈ 69.67 mm

Therefore, the maximum distance at which the dots can just be resolved by the camera is approximately 69.67 mm.