Question
Two identical phase connected waves, each of intensity $I _0$, are producing interference patterns. Write the value of the resultant intensity at the places of (i) constructive interference, (ii) destructive interference
✓
Answer
$I = I _1+ I _2+2 \sqrt{ I _1 I _2} \cos \phi$
Here, $\quad I _1= I _2= I _0$
(i) At constructive interference,
$\phi=2 n \pi \quad$ (where $n=0,1,2, \ldots$ )
So $\quad \cos \phi=1$
$\therefore \quad I = I _0+ I _0+2 \sqrt{ I _0 I _0}$
$= I _0+ I _0+2 I _0=4 I _0$
(ii) For destructive interference
$\phi=(2 n-1) \pi \quad($ Where $n=1,2, \ldots)$
Thus, $\quad \cos \phi=-I$
$\therefore \quad I = I _0+ I _0+2 \sqrt{ I _0 I _0}(-1)$
$=2 I _0-2 I _0=0$ (Zero)
Here, $\quad I _1= I _2= I _0$
(i) At constructive interference,
$\phi=2 n \pi \quad$ (where $n=0,1,2, \ldots$ )
So $\quad \cos \phi=1$
$\therefore \quad I = I _0+ I _0+2 \sqrt{ I _0 I _0}$
$= I _0+ I _0+2 I _0=4 I _0$
(ii) For destructive interference
$\phi=(2 n-1) \pi \quad($ Where $n=1,2, \ldots)$
Thus, $\quad \cos \phi=-I$
$\therefore \quad I = I _0+ I _0+2 \sqrt{ I _0 I _0}(-1)$
$=2 I _0-2 I _0=0$ (Zero)
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