Resolving pixels on a computer screen

[fubag]

[SOLVED] Resolving pixels on a computer screen

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 281 micrometers 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 lambda = 550 nanometers as a characteristic optical wavelength.

Homework Equations

I used Rayleigh's criterion which states (theta_minimum) = (lambda/a), where a is the width of the slit.

Then (theta_min) = (1.22(lambda))/D, where D is the aperture diameter.

The Attempt at a Solution

Using "a" as 281 micrometers, i tried solving and got D = .34mm which doesn't seem right nor work.

 

[alphysicist]

Hi fubag,

Your first expression (theta_minimum) = (lambda/a) is Rayleigh's criterion for a slit; your second expression is the same thing for a circular aperture.

Since your pupil is circular all you need to use is the circular aperture expression (theta_min) = (1.22(lambda))/D. Here theta_min is the angle that two adjacent pixels make in your field of view when you're a distance 1.3 meters away.

 

[fubag]

ok so I used theta_min = atan(281micrometers/1.3m) and solve for D?

This gave me D = 3.10 mm.

Is this correct?

 

[fubag]

ok I got that part

Part 2:

Assuming that the screen is sufficiently bright, at what distance can you no longer resolve two pixels on diagonally opposite corners of the screen, so that the entire screen looks like a single spot? Note that the size (0.360 meters) quoted for a monitor is the length of the diagonal.

do i take theta to be really really small and use D = 3.1mm to find that theta, or use the .360 meters as a part of atan for finding theta?

 

[fubag]

ok still stuck on this one...

I understand I am using Rayleigh's criterion formula where (theta_min) = lamda / a . So I think I have to take a = .36m, since it is now the width of the slit or something.

and using the properties that tan (a) = opposite/adjacent, i want to solve for opposite which will give me the distance, using adjacent as what though? this is bothering me.

 

[alphysicist]

ok still stuck on this one...

I understand I am using Rayleigh's criterion formula where (theta_min) = lamda / a . So I think I have to take a = .36m, since it is now the width of the slit or something.

and using the properties that tan (a) = opposite/adjacent, i want to solve for opposite which will give me the distance, using adjacent as what though? this is bothering me.

It looks to me like this is really the same type of problem as part a; just the distance between the pixels has changed. The diffraction would still be due to the pupil of the eye, so you'll need to use the circular version of Rayleigh's criterion formula.

 

[fubag]

ok so I find (theta_min) = (550 * 10^-9) / (.36m), using a as distance between the two points.

then solve for D = (1.22(550 * 10^-9)) / (theta_min) ?

This gives me .44m which makes no sense.

 

[fubag]

ok i actually solved it now

found my mistake

thanks!

 

[alphysicist]

You already know the diameter of the pupil D; it won't change. What you are looking for is L, the length from you to the screen. That means you also know theta_min.

What's the relationship between theta_min, the ditance to the screen, and the distance between the pixels?

 

[HeLLz aNgeL]

ok i actually solved it now

found my mistake

thanks!

i don't get it... how'd you solve for the distance ? arctan doesn't work ..

 

[minidee]

yeah i keep getting thetamin = 0.36/L
and answer 2029. what am I doing wrong?

 

[Redbelly98]

What is the value of thetamin (use the information in the problem statement)?

 

[minidee]

i got theta min is 1.774 x 10^-4 from 550x10-9/3.1mm I believe. What's my mistake?

 

[minidee]

would the theta min from 1.22 x 550 x 10-9 / 3.1mm be the right theta min then? I wuold get L=1660m for this theta min?

 

[minidee]

yep, the right answer from the machine said the 1.22 x 550 x 10-9/3.1 one was correct? Thanks!

 

[Redbelly98]

would the theta min from 1.22 x 550 x 10-9 / 3.1mm be the right theta min then? I wuold get L=1660m for this theta min?

Looks good.

... Thanks!

You're welcome