Question

In Young’s double slit experiment using light of wavelength $5000{{A}^{0}}$, what phase difference corresponds to the $11th$ dark fringe from the centre of the interference pattern?

Answer 1

In this question we have to use the concept of path difference obtained by dark and bright fringes on the screen when light from coherent sources interfere on screen and then from that path difference we will use the formula to convert it into phase difference so that the required result is obtained.

Complete step by step solution:
When a monochromatic light of particular wavelength is split by two slits in such a way two sources will be obtained from this single source and these sources are called coherent sources, which means that source having the same frequency and constant or zero phase difference. When waves from these sources interfere at any point on the screen then bright and dark fringes are obtained in an alternate manner on the screen and every point is having some path difference according to the fringe obtained on the screen.
In the given question we have to use that path difference to calculate the phase difference of $11th$ dark fringe.
So, path difference of dark fringe in young’s double slit experiment is represented by the relation:-
$p=\dfrac{(2n-1)\lambda }{2}$
Where
$n=1,2,3,4.......$
$n$ is representing the number of dark fringes.
$p$ is representing the path difference.
$\lambda$ is representing the wavelength of light used.
According to the question for $11th$ dark fringe.
Value of $n=11$
Put in above equation we get,
$p=\dfrac{21}{2}\lambda$
Since Path difference and phase difference are inter related to each others, it can be expressed as
$\phi$ is representing the phase difference

& \phi =\dfrac{2\pi }{\lambda }\times p \\\ & \phi =\dfrac{2\pi }{\lambda }\times \dfrac{21}{2}\times \lambda \\\ & \therefore \phi =21\pi \\\ \end{aligned}$$ So we can conclude that the phase difference corresponds to the $$11th$$ dark fringe from the centre of the interference pattern is $$21\pi $$. **Note:** Interference pattern can only be observed when two coherent sources are used; coherent sources are those sources which have same frequency and constant or zero phase difference for sustained interference pattern we have to take two sources of same amplitude so that we get perfect contrast in bright and dark fringes.