Question

Two coherent sources 'P' and 'Q' produce interference at point 'A' on the screen where there is a dark band which is formed between $4^{th}$ bright band and 5th bright band. The wavelength of light used is 6000A˚. The path difference between PA and QA is
A. $1.4 \times {10^{ - 4}}cm$
B. $2.7 \times {10^{ - 4}}cm$
C. $45 \times {10^{ - 4}}cm$
D. $6.2 \times {10^{ - 4}}cm$

Answer 1

Hint: As we all know that the sources which emit the light wavelength, frequency, and time are coherent sources. Coherent sources have zero phase difference or constant phase difference. For achieving coherence, the two sources of light must be kept close to each other.

Complete step by step solution:
As we all have studied that, the path difference is given by the term:
$\Delta x = \dfrac{{\lambda \delta }}{{2\pi }}$ …… (I)
Here $\Delta x$ is the path difference, $\lambda $ is the wavelength, and $\delta $ is the phase difference.
As we can see that, 4th dark fringe is formed after 4th bright fringe and the relation for phase difference is given by
$\delta = \left( {2n + 1} \right)\pi $ ……. (II)
Since we are talking about a dark fringe, we will put n=4 in equation (II) to find the value of path difference. Hence the equation (II) becomes,
$\delta = \left( {2 \times 4 + 1} \right)\pi $
$\delta = 9\pi $
Now we will substitute, $\delta = 9\pi $ and $\lambda = 6000\mathop A\limits^0 $ in the equation(I) to find the values of path difference between PA and QA, We will get,
$ \Rightarrow \Delta x = \dfrac{{\lambda \delta }}{{2\pi }}$
\[ \Rightarrow \Delta x = \dfrac{{6000\mathop A\limits^0 \times 9\pi }}{{2\pi }}\]
\[ \Rightarrow \Delta x = \dfrac{{6000\mathop A\limits^O \times 9\pi }}{{2\pi }}\]
\[ \Rightarrow \Delta x = \dfrac{{6000 \times {{10}^{ - 10}} \times 9\pi }}{{2\pi }}\]
After more simplifying the above equation, we will get the result as,
\[ \Rightarrow \Delta x = 3000 \times {10^{ - 10}} \times 9\]
\[\Rightarrow \Delta x = 2.7 \times {10^{ - 3}}cm\]

$\therefore$The path difference is \[2.7 \times {10^{ - 3}}cm\]. Hence, the correct option is (B).

Note:
As we have all noticed, when a film of oil floating in a puddle reflects light, a swirling mass of rainbow colors seem to magically appear. The cause of this phenomenon is the interference between light waves. A soap bubble floating in the air also reflects a variety of beautiful colors. Waves don't normally reflect when they strike one another. Instead, they combine. If the amplitudes of two waves have the same sign - either both positive or both negative, they will add together to form a wave with a larger amplitude.