Calculate Radius of Curvature for Newton's Rings - 5th & 15th Bright Ring

[The-Exiled]

Homework Statement

The diameter of the 5th and 15th bright ring formed by sodium light (wavelength = 589.3nm) are 2.303mm and 4.134mm respectively. Calculate the radius of curvature.

Homework Equations

DON'T ASSUME THAT THE RINGS ARE PERFECTLY CENTRED
(I assume that means that the radius of the rings can't be obtained by simply dividing the diameter by 2)

The Attempt at a Solution

I am not sure how to start finding the radius of the rings or if I need it at all? I know there is a $$\pi$$ phase change and hence the constructive interference is when
2 x [distance in air] = (m + 0.5)$$\lambda$$.

I've tried finding the ratios of diameters for some reason, just to see if it lead to anything... It didn't... Can anyone help me please?

Thank you very much

 

[mark726]

for your question. It's great to see that you are thinking critically about the problem and considering the possibility that the rings may not be perfectly centered. In order to calculate the radius of curvature, we will need to use the formula for the radius of a circle, which is r = d/2, where r is the radius and d is the diameter. However, since we cannot assume that the rings are perfectly centered, we will need to use the distance between the center of the rings and the center of the circle, which we will call x.

Using the formula you mentioned for constructive interference, 2x = (m + 0.5)λ, we can solve for x by plugging in the known values for the wavelength and the order of the ring (m). Once we have the value for x, we can use the Pythagorean theorem to calculate the radius of the circle, which is given by the equation r = √(x^2 + (d/2)^2).

In this case, we will need to calculate the radius for both the 5th and 15th ring, using the respective diameters given. Once we have the two values for the radius, we can use them to calculate the radius of curvature by taking the average of the two values. This will give us a more accurate estimate, taking into account any potential errors in the measurement of the ring diameters.

I hope this helps and good luck with your calculations!