Newton's Rings and Thin Film Interference

[Abu]

Homework Statement

A total of 31 bright and 31 dark Newton's rings (not counting the dark spot at the center) are observed when 550-nm light falls normally on a planoconvex lens resting on a flat glass surface. How much thicker is the center than the edges?

Homework Equations

2t = mλ
2t = (m+1/2)λ

The Attempt at a Solution

The above image is just one I found online. As for my attempt:
Since the incident ray is being reflected off of a surface where it would have gone faster had it not been reflected, there is no phase shift. However, when it reflects off of the flat glass, it is being reflected off of a surface where it would have gone slower had it not been reflected, thus giving a phase shift.

That means that the formula I will be using is:
2t = mλ for destructive interference
2t = (m+1/2) for constructive interference.

As it is asking how much thicker the center is than the end, my first instinct is to find the thickness of the center, than find the thickness at the very edge, and find the difference.

Since the center is destructive however, the formula used is 2t = mλ
The center means that m = 0
That means that the thickness of the air film between the curved surface is zero (which I think makes sense because there is no room for air at the center)

The furthermost point of the Newton rings is destructive interference (NOT SURE ABOUT THIS) because the last ring would have color, and beyond that there is no more color.

So I am tempted to use this formula:
2t = mλ for destructive interference
So...
2t = 31(550)
t = 8525 nm

Can someone tell me if my reasoning is right? Specifically how I assume the furthermost point is destructive interference and also how I assume that the thickness in the center is zero. Also, why does the question say how much thicker is the center than the edges when in reality the center has zero thickness (value for t was zero)

Thank you for your time.

 

[haruspex]

how much thicker is the center than the edges when in reality the center has zero thickness

The question is asking about the thickness of the lens. The zero thickness you found is for the airspace under the lens.

 

[Abu]

The question is asking about the thickness of the lens. The zero thickness you found is for the airspace under the lens.

The answer for the question is 8.5 micrometers in the textbook. Since I got 8525 nm from my attempt, this makes 8.525 micrometers, which is the thickness of the airspace under the lens at the edges. So that is why I found it confusing why they said how much thicker is the center than the edges. But I do realize that the width of the lens in the center is larger than at the edges, so it makes me wonder if there is a typo in the textbook question or if I am misunderstanding something

 

[haruspex]

The answer for the question is 8.5 micrometers in the textbook. Since I got 8525 nm from my attempt, this makes 8.525 micrometers, which is the thickness of the airspace under the lens at the edges. So that is why I found it confusing why they said how much thicker is the center than the edges. But I do realize that the width of the lens in the center is larger than at the edges, so it makes me wonder if there is a typo in the textbook question or if I am misunderstanding something

I am not sure whether you understood what I wrote.
There is no typo. When the question asks how much thicker at the centre than at the edge it is referring to the lens thickness, not the airspace thickness. It is not really even ambiguous. If it meant the thickness of the airspace it would have said so, or, more likely, it would have asked for the depth of the airspace. "Thickness" suggests something solid.

 

[Abu]

I am not sure whether you understood what I wrote.
There is no typo. When the question asks how much thicker at the centre than at the edge it is referring to the lens thickness, not the airspace thickness. It is not really even ambiguous. If it meant the thickness of the airspace it would have said so, or, more likely, it would have asked for the depth of the airspace. "Thickness" suggests something solid.

Okay, sorry sir. I am only expected to be using 2t = mλ and 2t = (m+1/2) . I realize that the question is asking for the thickness of the lens. But first, am I wrong in saying that t in the formula 2t = mλ means the thickness of the airspace? If t in the formula 2t = mλ is in fact the thickness of the airspace, then I have no idea how to find the thickness of the lens. All I know is that the final answer for the question in the textbook answer sheet is 8.5 micrometers. The final answer I got was 8.525 micrometers, but it was for the thickness of the airspace underneath the edge.

Thank you for your patience.

 

[haruspex]

If t in the formula 2t = mλ is in fact the thickness of the airspace, then I have no idea how to find the thickness of the lens.

In relation to the diameter of the whole lens, where do you think the 31^st dark ring is?

 

[Abu]

In relation to the diameter of the whole lens, where do you think the 31^st dark ring is?

The 31st dark ring is at the edge of the lens. It is the furthermost point from the center.

 

[haruspex]

The 31st dark ring is at the edge of the lens. It is the furthermost point from the center.

Right, so what is the thickness of the lens there?
How much deeper is the airspace at the edge than in the centre?
So how much thicker is the lens in the centre than at the edge?

 

[Abu]

Right, so what is the thickness of the lens there?
How much deeper is the airspace at the edge than in the centre?
So how much thicker is the lens in the centre than at the edge?

I just realized. The airspace at the edge is the thickness of the lens at the center, because the airspace at the center is zero. If that is correct, than holy cow I am either extremely dumb or just really tired haha.

Besides that, were my other assumptions in my original attempt at the question correct?

Thank you sir.

 

[haruspex]

If that is correct, than holy cow I am ... really tired haha.

Yes.