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**Newtons Rings Calculations - Values Not Working Out**

I'm having trouble with a Newtons Rings experiment to determine the refractive index of water. I'm using a sodium light source and a Vernier scale travelling microscope to measure the radius of the

__bright__circles (I don't know if measuring the dark circles will make a difference, though I have results for both). The problem arises not with finding the rings through the mircoscope, I can see the pattern clearly with a dark circle at the centre. The problem arises when I try to calculate the refractive index of

__air__for my control experiment.

Using the equations, y[itex]^{2}[/itex] = mRλ/n where y[itex]^{2}[/itex] = the radius of the bright rings, m = the m[itex]^{th}[/itex] ring, R = the radius of the underside of the lens and n = refractive index.

Rearranging the formula: n = mRλ/y[itex]^{2}[/itex]

**Results**

Centre of pattern = 3.21mm (on Vernier Scale)

1st ring = 3.21mm ∴ y[itex]^{2}[/itex] = 0.02mm

2nd ring = 3.24mm ∴ y[itex]^{2}[/itex] = 0.05mm

3rd ring = 3.25mm ∴ y[itex]^{2}[/itex] = 0.06mm

(I have results for 10 rings, which I can post if it is helpful)

R = 75.23x10[itex]^{-3}[/itex] (using Pythagoras' Theorm)

Using the above formula: n = mRλ/y[itex]^{2}[/itex]

n = 1 x 75.23x10[itex]^{-3}[/itex] x 589.3x10[itex]^{-9} / 0.00002^{2}[/itex]

=

__110.8325975__I realise that this number is almost correct. I have the right digits just to the wrong order. However, when I did this calculation with the values for the second ring (m = 2, y[itex]^{2}[/itex] = 0.05[itex]^{2}[/itex]) the answer I got was

__35.4664312__.

I've tried everything I can think of and I cannot find any other formula that uses refractive index and wavelength. The only thing I can come up with as to why this does not work is because the paper in which I got this formula from is measuring water (which I will be ultimately, though I am measuring air for the moment) and that the formula only works for water. However, I very much doubt this as the only variable to change is the presence of water between the optical flat and lens.

Any suggestions will be greatly appreciated and I am more than willing to provide any extra data that I have if needed.

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