Assume that the dots in the painting are separated by 1.0 mm and that the wavelength of the light is vacuum= 500 nm. Find the distance at which the dots can just be resolved by (a) the eye and (b) the camera.
I know that the resolving equation is (angle)min = 1.22(wavelength) / D
From the text, D(eye)=2.5 mm and D(camera)= 25mm.
I can solve for the (angle)min, but don't know how to find the distance.
To find the distance at which the dots can just be resolved by the eye and the camera, we can use the resolving equation you provided:
(angle)min = 1.22(wavelength) / D
where (angle)min is the smallest resolvable angle, wavelength is the wavelength of light, and D is the distance between the dots.
(a) Resolving distance for the eye:
Given D(eye) = 2.5 mm and wavelength = 500 nm, we can substitute these values into the equation to solve for (angle)min:
(angle)min = 1.22 * (500 nm) / (2.5 mm)
Make sure to convert all units to a consistent system, such as meters. In this case, convert 2.5 mm to meters by dividing by 1000:
(angle)min = 1.22 * (500 * 10^-9 m) / (2.5 * 10^-3 m)
Calculating this expression gives us the value of (angle)min in radians. To find the corresponding distance, we can use the relationship:
(distance) = D(eye) / tan(angle)
where (distance) is the distance at which the dots can just be resolved. Substitute the values of D(eye) and (angle)min into this equation to get the final answer.
(b) Resolving distance for the camera:
Similarly, given D(camera) = 25 mm, we can substitute this value into the equation to solve for (angle)min:
(angle)min = 1.22 * (500 nm) / (25 * 10^-3 m)
Again, convert all units to a consistent system, such as meters, and then calculate the expression. Finally, use the relationship (distance) = D(camera) / tan(angle) to find the distance at which the dots can just be resolved for the camera.
Note: While this approach provides an estimate, it is important to consider that the actual resolving power of the eye and the camera can vary depending on factors like the clarity of the lens, the lighting conditions, and individual differences in visual acuity.