Question

What is the path difference formula?

Answer 1

Hint: As a first step you could define the path difference and also explain the consequences of path difference that exists between two waves. Then you could give the formula that is commonly used. Remember to introduce the other terms that are present in that formula. Also, given the path difference for the two types of interferences caused.

Complete answer:
In the question we are asked to give the path difference formula. Let us first define the path difference. Just as the name says, it is indeed the difference in path traversed by two waves that are meeting a particular point. The slight difference in this path traversed does have a huge impact. The waves can cause constructive and destructive interference depending on this value of path difference. Real life examples that you may be familiar with are soap bubble and oil slick.
We commonly use the following formula for finding path difference which is in terms of phase difference. Phase difference is the difference between the phase angles of the two waves under consideration. The relation could be given as,
$\mathrm{\Delta }x={\displaystyle \frac{\lambda }{2\pi }}\mathrm{\Delta }\varphi$
Here, $\mathrm{\Delta }x$ = path difference
$\mathrm{\Delta }\varphi$ = phase difference
$\lambda$ = wavelength

Note:
We have mentioned in the solution about the constructive and destructive interference that can result due to the path difference that exists between the waves. Now for the destructive interference the path difference would be half integer number of wavelength, that is,
$\mathrm{\Delta }x=(n+{\displaystyle \frac{1}{2}})\lambda$
Now for the constructive interference, the path difference would be integer number of wavelengths, that is,
$\mathrm{\Delta }x=n\lambda$