Newtons Rings: Calculating Refractive Index

[cgraystone]

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

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


Results
Centre of pattern = 3.21mm (on Vernier Scale)

1st ring = 3.21mm ∴ y^{2} = 0.02mm
2nd ring = 3.24mm ∴ y^{2} = 0.05mm
3rd ring = 3.25mm ∴ y^{2} = 0.06mm
(I have results for 10 rings, which I can post if it is helpful)

R = 75.23x10^{-3} (using Pythagoras' Theorm)

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

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

= 110.8325975​

I realize 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^{2} = 0.05^{2}) 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.

 

[Basic_Physics]

The diameters, D, of the dark rings (in air) are related to the wavelength, λ, and radius of curvature, R, of the lens (the radius of the spherical ball of which the lens is a part) is given by

λ = (D²_(n+m) - D²_n)/4mR

 

[cgraystone]

λ = (D²_(n+m) - D²_n)/4mR

Thanks very much for replying. I'm a little confused though; do you mean to take D²_(n+m) and D²_n as two separate values? For this project I need to be able to have n as the subject of the equation.

 

[Basic_Physics]

You mention a lot of "objectives" for this experiment. So I thought it good to start somewhere/anywhere. This might not be what you want since it seems your first objective is to determine the refractive index of air? Although this formula might get you there.
The formula takes the difference between the square of the diameters of two dark rings.
D_n is the diameter of a smaller dark ring and D_(n+m) is the diameter of a larger dark ring -the m^th one outside of D_n. It is best to start at one side of the rings with the traveling microscope and just keep on traveling in one direction while taking measurements of the encountered rings as you go. This eliminates backlash in the traveling microscope.

Your formula is correct (for the dark rings). It just seem that the value you get for the refractive index will have a large uncertainty due to the fact that you have only one significant digit to work with for the radius square. So rather try this formula:
D^{2}_{n}=\frac{4n\lambda R}{\mu}
where μ is the index of refraction. You might also get better accuracy if you rather use the difference formula, but your results will depend on your measuring technique. If you did not measure correctly your errors will be large and accuracy will be severely influenced.