When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh’s criterion. Take the pupil of the eye to be 2.9 mm and the distance from the paper to the eye of 31 cm; find the minimum separation of two dots such that they cannot be resolved.
I tried using Rayleigh's criterion to solve for the angle. I then converted it from radians to degrees and used it in the equation for resolving power, but did not get the right answer. (I got 14.50 cm)
To find the minimum separation of two dots such that they cannot be resolved using Rayleigh's criterion, we need to understand the formula and make sure we're using the correct inputs.
Rayleigh's criterion states that two points or dots can be just resolved when the angle between them (θ) is equal to or greater than the minimum resolvable angle (θ_min), which can be calculated using the formula:
θ_min = 1.22 * (λ / D)
where:
- θ_min is the minimum resolvable angle (in radians)
- λ is the wavelength of light
- D is the diameter of the pupil of the eye
First, we need to convert the pupil diameter from millimeters to meters:
D = 2.9 mm = 0.0029 m
Next, we need to calculate the wavelength of light used in the laser printer. Assume it uses visible light with a wavelength of 550 nm (nanometers). We also need to convert this to meters:
λ = 550 nm = 550 * 10^-9 m = 0.00000055 m
Now, we can substitute the values into the formula for θ_min:
θ_min = 1.22 * (0.00000055 m / 0.0029 m)
Calculating this, we get:
θ_min ≈ 2.31 * 10^-4 radians
To convert this into degrees, we use the conversion factor: 1 radian = 57.3 degrees.
θ_min ≈ 2.31 * 10^-4 radians * 57.3 degrees/radian
θ_min ≈ 0.0132 degrees
Now that we have the minimum resolvable angle, we can use simple trigonometry to find the minimum separation of two dots at a given distance from the eye. In this case, the distance from the paper to the eye is 31 cm.
Using the formula:
Minimum Separation = distance * tan(θ_min)
Minimum Separation = 31 cm * tan(0.0132 degrees)
Calculating this, we have:
Minimum Separation ≈ 0.005 cm
Therefore, the minimum separation of two dots that cannot be resolved is approximately 0.005 cm.
It seems there was an error in your calculation, resulting in an incorrect answer of 14.50 cm. Going through the steps again with the correct inputs should lead you to the correct answer of 0.005 cm.
To find the minimum separation of two dots such that they cannot be resolved using Rayleigh's criterion, you need to calculate the angular resolution of the human eye and then convert it into a linear separation on the page.
1. Start by finding the angular resolution using Rayleigh's criterion:
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.
2. Convert the angular resolution from radians to degrees:
θ_degrees = θ * (180 / π)
3. Calculate the linear separation on the page using the formula:
separation = distance * tan(θ)
where separation is the minimum separation of the dots on the page, and distance is the distance from the paper to the eye.
Let's calculate the minimum separation in the steps mentioned above:
Step 1:
λ = wavelength of light (typically taken as 550 nm)
D = diameter of the pupil (2.9 mm)
θ = 1.22 * (550 nm / 2.9 mm)
Step 2:
θ_degrees = θ * (180 / π)
Step 3:
distance = 31 cm (0.31 m)
separation = 0.31 m * tan(θ)
Now you can perform the calculations and find the correct value for the minimum separation.