Solving Aperture Question Homework: Diameter of Pupil

[toesockshoe]

Homework Statement

A standard 14.16-inch (0.360-meter) computer monitor is 1024 pixels wide and 768 pixels tall. Each pixel is a square approximately 281micrometers on each side. Up close, you can see the individual pixels, but from a distance they appear to blend together and form the image on the screen.

If the maximum distance between the screen and your eyes at which you can just barely resolve two adjacent pixels is 1.30 meters, what is the effective diameter d of your pupil? Assume that the resolvability is diffraction-limited. Furthermore, use λ=550 nanometers as a characteristic optical wavelength.

Homework Equations

$$sin(\theta) = \frac{1.22* \lambda}{D}$$

The Attempt at a Solution

using small angle formula, you can assume $$sin(\theta) = \theta$$ So, $$D = \frac {1.22 \lambda}{\theta}$$. $$\theta = \frac{281}{10^6}$$ .. AND $$\lambda = \frac{550}{10^9}$$. Plugging all these values in, I get D=2.387 mm which is incorrect according to the homework checker. Where am I going wrong?

 

[TSny]

$\theta = \frac{281}{10^6}$

Hello. Can you explain why the above expression represents θ? If you were to include units with your numbers, would you get the correct units for θ?

 

[toesockshoe]

Hello. Can you explain why the above expression represents θ? If you were to include units with your numbers, would you get the correct units for θ?

It is 281micrometers/10^6 micrometers. This would be dimensionless which is a valid argument for theta.

 

[toesockshoe]

It is equivalently 281 times 10^-6 m/1m.

 

[TSny]

It is equivalently 281 times 10^-6 m/1m.

Why the 1 m in the denominator?

 

[toesockshoe]

Why the 1 m in the denominator?

should it be 1.3m in the denominator instead? because tantheta is approx sin theta for small angles? ... i still don't get right answer.

 

[TSny]

Forget the numbers for a moment. Let's make sure we understand the concept. Can you describe in words how to find θ?

 

[toesockshoe]

Forget the numbers for a moment. Let's make sure we understand the concept. Can you describe in words how to find θ?

yeah... you can use tan^-1(opposite/adjacent). opp is the distance between 2 pixels, and adj distance is the distance from you to the screen (which is 1.3). we have the opp distance and adj distance so we can find theta. units, cancel out because you are dividing meters/meters... and you have an acceptable angle measure.

 

[TSny]

OK, that will work for small angles. For small angles you can go on and approximate d/L by θ (in radians), where d is distance between pixels and L is distance from eye to pixels. So, yes, you should be dividing by 1.3 m.

I'm not sure why you are not getting the right answer now. What are you getting for D?

 

[toesockshoe]

OK, that will work for small angles. For small angles you can go on and approximate d/L by θ (in radians), where d is distance between pixels and L is distance from eye to pixels. So, yes, you should be dividing by 1.3 m.

I'm not sure why you are not getting the right answer now. What are you getting for D?

I got it. thank you.

 

[TSny]

Good work.