Question
What are coherent sources ? Explain Constructive interference using proper example.
✓
Answer
→Consider two needles $S _1$ and $S _2$ moving periodically up and down in an identical fashion in a trough of water. (fig. (a))
→Consequently two ripples (/water waves) are created in water. When the two waves superimpose at a particular point, the phase difference between the displacements produced by each of the waves does not change with time.
→When this happens, the two sources are said to be coherent sources.
→Consider a point P for which,
$S _1 P = S _2 P$
→Since distances are equal, waves from $S _1$ and $S _2$ will take the same time to travel to point $P$ and waves that emanate from $S_1$ and $S_2$ in phase will also arrive, at point $P$ in phase.
→Displacement produced by the source $S_1$ at point P is given by,
$y_1=a \cos \omega t$
→Displacement produced by the source $S _2$ at point P is given by,
$y_2=a \cos \omega t$
→As per superposition principle,
$\begin{aligned}
y & =y_1+y_2 \\
\therefore y & =a \cos \omega t+a \cos \omega t \\
\therefore y & =2 a \cos \omega t
\end{aligned}$
→The intensity of a wave is proportional to the square of the amplitude.
So, the resultant intensity is given by :
$I=4 I_0$
where $I _0$ represents the intensity produced by each one of the individual sources.
→Here, the intensity at point P is maximum, which is known as the constructive interference.
→Now, as shown in fig. (c), consider a point $Q$ for which,
$S _2 Q - S _1 Q =2 \lambda$
→The waves emanating from $S_1$ will arrive exactly two cycles earlier, than the waves from $S _2$ and will be in phase. Path difference of $2 \lambda$ corresponds to a phase difference of $4 \pi rad$. Hence, the wave coming from $S _2$ will be late in phase by $4 \pi$ radian.
→If the displacement produced by $S _1$ is
$y_1=a \cos \omega t$
→Then the displacement produced by $S _2$ will be,
$\begin{aligned}
y_2 & =a \cos (\omega t-4 \pi) \\
\therefore \quad y_2 & =a \cos \omega t
\end{aligned}$
→Net displacement at point Q ,
$\begin{aligned}
y & =y_1+y_2 \\
\therefore y & =a \cos \omega t+a \cos \omega t \\
\therefore y & =2 a \cos \omega t
\end{aligned}$
→The two displacements are in phase once again, and the intensity once again will be $4 I _0$ giving rise to the Constructive interference.
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