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I came across this question

A lightsource of diameter 10cm is placed 2m underneath the surface of a pool, a person 5m away from the edge of the pool saw a circle of light emitting from the lamp. Calculate the maximum diameter of the circle of light seen by the person. (Use refractive index = 1.33)

I figured this would be something similar to that of Snell's Law and drew the diagram below: enter image description here

The answer is indeed about 4.66m, however I am quite puzzled as to how the person could see the entire circular shape of light, given that some rays are travelling away from the person?

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  • $\begingroup$ I think it’s not necessary that the circle of light seen by the person is exactly centred on the source of light like it is in your diagram. The circle of light seen by the person will be offset towards the person. The farthest part of the circle seen by the person will actually be in between the source of light and the user. $\endgroup$
    – Fogmeister
    Dec 24, 2021 at 7:19
  • $\begingroup$ Hint : $\:2.28m = 2m \cdot \tan[\arcsin(1/n)]$. $\endgroup$
    – Frobenius
    Dec 24, 2021 at 14:07

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Edit

This question is badly written and (if they gave the correct answer as 4.66m as you put in your question) then not physically accurate at all.

The question asks...

A lightsource of diameter 10cm is placed 2m underneath the surface of a pool, a person 5m away from the edge of the pool saw a circle of light emitting from the lamp. Calculate the maximum diameter of the circle of light seen by the person. (Use refractive index = 1.33)

This is a really badly formed question.

What it is asking as written and what it is trying to get you to calculate are two vastly different things.

What it is trying to get you to calculate...

...is the diameter of the circle of light that will escape the water and not undergo total internal reflection.

This is indeed about 4.66m.

But... the person plays absolutely no part in this at all. The introduction of the person in this question throws this off completely and is a complete red herring.

Also... the fact that the light source is 10cm in diameter is completely irrelevant too. I mean... it slightly (very slightly) will affect the diameter of the circle you calculate. But not by much.

What the question is asking as written...

... is to calculate the apparent size of the light source as seen by the person.

To be able to answer this question you would need to know the size of the light source, the height of the person, the distance of the light source from the edge of the pool, etc...

This will be approximately 10cm. Maybe a few percent bigger than 10cm.

How the question should have been written...

A point lightsource is placed 2m underneath the surface of a pool. Calculate the maximum diameter of the circle of light that is emitted out of the pool. (Use refractive index = 1.33)

Conclusion

I'm not surprised you were confused by this and I apologise for not spotting this ambiguity earlier. It was only when I was trying to draw this myself that I realised what was going on.

If you were given this by a teacher you should mention to them that the question needs rewriting to remove...

  1. the person
  2. the size of the light source (it should be a point light source)
  3. the distance between the light source and the edge of the pool

... and also that the question needs rewriting.

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  • $\begingroup$ but if the diagram is like what you drew, then the person will see a semi-circle of light on the surface then? $\endgroup$
    – tangolin
    Dec 25, 2021 at 4:00
  • $\begingroup$ By your diagram in that case won't the diameter be just half of 4.66m? $\endgroup$
    – tangolin
    Dec 26, 2021 at 1:45
  • $\begingroup$ I will do that. Thank you for your help! $\endgroup$
    – tangolin
    Dec 26, 2021 at 9:30
  • $\begingroup$ Not a problem. I’m going to try that also. I’d suggest drawing rays of light from the person to both edges of the pool and at regular intervals (every 20cm) across the pool. To start with. And refract all of them. $\endgroup$
    – Fogmeister
    Dec 26, 2021 at 9:32
  • $\begingroup$ @tangolin I have edited my question. I would suggest your teacher does the same to their question. 😅 I am not surprised you were confused by this. It took me to start diagramming it myself to realise that the question is really badly worded. $\endgroup$
    – Fogmeister
    Dec 26, 2021 at 10:40

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