Two small identical speakers are connected in phase to the same s… - Vidyadip

MCQ

Two small identical speakers are connected in phase to the same source. The speakers are $3 \,m$ apart and at ear level. An observer stands at $P, 4 \,m$ in front of one speaker as shown alongside. The sound she hears is least intense when the wavelength is $\lambda_1$ and most intense when the wavelength is $\lambda_2$. Then, the possible values of $\lambda_1$ and $\lambda_2$ are

A
$\lambda_1=1 \,m$ and $\lambda_2=2 \,m$
B
$\lambda_1=4 \,m$ and $\lambda_2=3 \,m$
✓
$\lambda_1=2 \,m$ and $\lambda_2=1 \,m$
D
$\lambda_1=0.5 \,m$ and $\lambda_2=025 \,m$

✓

Answer

Correct option: C.

$\lambda_1=2 \,m$ and $\lambda_2=1 \,m$

c
(c) Path difference of sounds from speakers $A$ and $B$ is $1 \,m (=5 \,m -4 \,m )$.

For least intense sound, $1 \,m$ can cause a destructive interference. So, path difference is an odd multiple of half wavelength.

$\Rightarrow \Delta L=(2 n+1) \frac{\lambda_1}{2}$

$\Rightarrow 1 \,m =\frac{\lambda_1, 3 \lambda_1, 5 \lambda_1}{2}, \frac{2}{2}, \text { etc. }$

Above condition is satisfied, when $\lambda_1=2 \,m$.

And for most intense sound, path difference of $1 m$ can cause a constructive interference. So, path difference must be a whole number multiple of one wavelength.

$\Rightarrow \Delta L=n \lambda_2$

$\Rightarrow 1 \,m =\lambda_2, 2 \lambda_2, 3 \lambda_2, \ldots, \text { etc. }$

Above condition is satisf ied with $\lambda_2=1 \,m$.

So, correct option is $(c)$.

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