Optics, resolution problem sheet

[hayze2728]

Hey guys, I've been set this problem sheet for optics and I haven't got a clue how to complete it, the course was really rushed and all of us have been left confused, so any help would be greatly appreciated.

1. A man is stranded. He needs assistance and his mobile battery is dead. How large does he need to write the letters 'Help!' in the saad for the team back at CTU to read from the images taken by the spy satellite 250km above him if its mirror is 2.5m in diameter, has a focal ratio of f/4, uses a filter centred at 500nm and works at the diffraction limit? (6 marks)

Consider the physical size of the diffraction limited image at the focus assuming the angular magnification of the image is unity(?). If the telescope was limited by the physical size of each element of the charge-coupled device at the telescope focus, how much larger would he have to write the letters? (2 marks)



2. I'm assuming Rayleigh''s Criterion: theta min = 1.22 (lambda/D) is one to use.



3. Like I said we've tried many things and would just like to know how to tackle the problem.

Thanks!

 

[Nate810]

To answer the first part of your question, you need to use the Rayleigh criterion. This states that the minimum angle of resolution (theta min) is equal to 1.22 times the wavelength of the light divided by the diameter of the telescope objective (D). Using this equation, you can calculate the minimum angle of resolution for the given telescope: theta min = 1.22 x (500 nm / 2.5 m) = 0.0968 radians. To calculate the size of the letters 'Help!' that the man needs to write for them to be seen from 250 km away, you need to multiply the distance to the telescope and the minimum angle of resolution: size of letters = 250 km x 0.0968 radians = 24.2 m. For the second part of the question, you need to consider the physical size of the diffraction limited image at the focus, assuming the angular magnification of the image is unity. In other words, you need to calculate the physical size of the image created by the telescope. To do this, you need to use the equation physical size = focal length x (angular size / angular magnification). Using this equation, you can calculate the physical size of the image: physical size = 10 m x (0.0968 radians / 1) = 10 m. Therefore, if the telescope was limited by the physical size of each element of the charge-coupled device at the telescope focus, the man would have to write the letters 'Help!' at a size of 10 m.

 

[ogb p]

Hello there,

I can understand your confusion with this problem sheet. Optics can be a complex subject, especially when it comes to resolving objects at a distance. Here are some steps you can follow to tackle this problem:

1. Start by understanding the given information. The man is stranded and needs assistance, and he wants to write the letters 'Help!' in the sand for the team back at CTU to see from a spy satellite. The satellite is 250km above him and has a mirror diameter of 2.5m, a focal ratio of f/4, and uses a filter centered at 500nm. It also works at the diffraction limit.

2. Determine the angular size of the letters 'Help!' at the distance of 250km. This can be calculated using the formula: theta = size/distance. The size of the letters can be assumed to be a few centimeters, and the distance is 250km.

3. Use Rayleigh's Criterion (theta min = 1.22 (lambda/D)) to calculate the minimum angular size that can be resolved by the telescope. Here, lambda is the wavelength of light (500nm) and D is the diameter of the telescope's mirror (2.5m).

4. Compare the angular size of the letters 'Help!' with the minimum angular size that can be resolved. If the letters are larger than the minimum angular size, they should be visible from the satellite.

5. Now, to answer the second part of the question, we need to consider the physical size of the diffraction limited image at the focus. This can be calculated using the formula: size = (1.22 lambda*f)/D. Here, f is the focal ratio (f/4) and D is the diameter of the telescope's mirror (2.5m).

6. Finally, to determine how much larger the letters need to be written, we need to consider the physical size of each element of the charge-coupled device (CCD) at the telescope focus. This can vary depending on the specific CCD used, so you may need to do some research on that.

I hope this helps you in tackling the problem sheet. Remember to always start by understanding the given information and using relevant formulas and concepts to solve the problem. If you still have trouble, don't hesitate to reach out for further assistance. Good luck!