Question

vi. thinking questions

• in the experiment, if the vertical line does not pass through the center of the ring, the measured value is the chord length rather than the diameter, does it effect the result?
• can the experimental results of newtons rings be treated graphically? if yes, how?

Explanation:

First Question:

For the first question about Newton's rings experiment: In the Newton's rings formula (e.g., for the radius of curvature \( R=\frac{r_{m}^{2}-r_{n}^{2}}{(m - n)\lambda} \), where \( r \) is the radius of the ring, \( m,n \) are orders, \( \lambda \) is wavelength), if we use chord length \( l \) instead of diameter (or radius), by the Pythagorean theorem, \( r^{2}=(\frac{l}{2})^{2}+d^{2} \), but when calculating \( r_{m}^{2}-r_{n}^{2} \), the \( d^{2} \) terms cancel out (\( r_{m}^{2}-r_{n}^{2}=(\frac{l_{m}}{2})^{2}+d^{2}-[(\frac{l_{n}}{2})^{2}+d^{2}]=\frac{l_{m}^{2}-l_{n}^{2}}{4} \)). So the difference \( r_{m}^{2}-r_{n}^{2} \) (or equivalent with chord lengths) remains the same, so using chord length instead of diameter does not affect the result for \( R \) calculation.

Yes, Newton's rings experimental results can be treated graphically. For example, we can plot a graph of \( r^{2} \) (or \( l^{2}/4 \) where \( l \) is chord length) against the order \( m \) of the rings. From the Newton's rings formula \( r_{m}^{2}=mR\lambda \) (for bright rings, approximate), this should be a straight line with slope \( R\lambda \). By finding the slope of the best - fit line to the experimental data points (\( (m, r_{m}^{2}) \) or \( (m, l_{m}^{2}/4) \)), we can calculate \( R \) (since \( R=\text{slope}/\lambda \)) or analyze the linearity to check the validity of the theory. Another way is to plot the number of rings vs. the square of the ring radius (or chord - length - related quantity) to see the consistency with the theoretical linear relationship.

Answer:

Using chord length instead of diameter does not affect the result. Because when calculating the difference of the squares of the radii (or related terms for Newton's rings formula), the non - central distance term cancels out, so the final result for the radius of curvature (or other derived quantities) remains unchanged.

Second Question: