In a double-slit experiment, the optical path difference between … - Vidyadip

Question

In a double-slit experiment, the optical path difference between the waves coming from two coherent sources at a point $P$ on one side of the central bright band is $7.5 \times 10^6 m$ and that at a point $Q$ on the other side of the central bright band is $1.8 \times 10^6 m$. How many bright and dark bands are observed between points $P$ and $Q$ if the wavelength of light used is $6 \times$ $10^7 m$ ?

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Answer

Data : $\Delta l _1=7.5 \times 10^{-6} m , \Delta l _2=1.8 \times 10^{-6} m \lambda=6 \times 10^{-7} m$
For point $P :$ Let $p \frac{\lambda}{2}=\Delta l _1$
$
\therefore p=\frac{2 \Delta l_1}{\lambda}=\frac{2 \times 7.5 \times 10^{-6}}{6 \times 10^{-7}}=\frac{150}{6}=25
$
The path difference $\Delta l _1$ is an odd integral multiple of $\lambda / 2: \Delta l _1=(2 m -$
1) $\frac{\lambda}{2}$, where $m$ is an integer,
$\therefore 2 m -1=25 \therefore m =13$
$\therefore$ Point $P$ is at the centre of the 13 th dark band.
For point Q :
Let $q \frac{\lambda}{2}=\Delta l _2$
$
\therefore q=\frac{2 \Delta l_2}{\lambda}=\frac{2 \times 1.8 \times 10^{-6}}{6 \times 10^{-7}}=\frac{36}{6}=6
$
The path difference $\Delta I _2$ is an even integral multiple of $\frac{\lambda}{2}: \Delta l _2=(2 n ) \frac{\lambda}{2}$,
where $n$ is an integer
$
\therefore 2 n =6 \therefore n =3
$
$\therefore$ Point $Q$ is at the centre of the 3rd bright band. Between points $P$ and $Q$, excluding the respective bands at $P$ and $Q$, the number of dark bands $=12+3=15$ and the number of bright bands (including the central bright band) $=12+2+1=15$

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