Question

In Young's double slit experiment, the path difference between two interfering waves at a point on the screen is \[\dfrac{{5\lambda }}{2}\]​,\[\lambda \] being wavelength of the light used. The ______ dark fringe will lie at this point. Fill in the blank with appropriate answer:

Answer 1

Hint:In order to solve this question, we should be aware of the young’s double slit experiment. The question is asked about the condition of minimum intensity. So, we will use the formula which involves path difference and the condition of minimum intensity.

Formula Used:
The path difference for ${n^{th}}$ dark fringe would be given by
$\Delta x = (2n - 1)\dfrac{\lambda }{2}$
Here, $\Delta x$ is the path difference
$n$ is the order of fringe
\[\lambda \] is the wavelength of light used while conducting the experiment.

Complete step by step solution:
According to the question,
Path difference $\Delta x = \dfrac{{5\lambda }}{2}$ ……. (Equation I)
Now, for ${n^{th}}$ dark fringe $\Delta x = (2n - 1)\dfrac{\lambda }{2}$ …….. (Equation II)
As from Equation I and II, we can say that both are the values of $\Delta x$ .
Hence, we can equate both the equations.
So, on comparing Equation (I) and (II)
We have,
$(2n - 1)\dfrac{\lambda }{2} = \dfrac{{5\lambda }}{2}$
Cancelling \[\dfrac{\lambda }{2}\] from both the sides we have,
$(2n - 1) = 5$
On solving for the value of $n$ we get,
$n = 3$
Hence, the answer is 3.

Note:We know that Young’s double slit experiment was done to understand and prove the wave nature of light. Young’s double slit experiment uses two coherent sources of light, that are placed at a small distance apart, usually, of a few orders of magnitude greater than the wavelength of the light used for the experiment. This experiment was first done by Thomas Young and that is why it is known as the Young’s experiment and it belongs to a general class of double slit experiments in which a light wave split into two separate waves that later combine into a single light wave. Changes in the path length of both of the light waves result in a phase shift, which creates an interference pattern. The wave nature of light causes the waves passing through the slit to interfere which produce bright and dark bands or fringes on the screen and this result is not expected if the light consisted of classical particles, which proves the wave nature of light.