Question

Ratio of maximum intensity and minimum intensity in interface is 25:1. Hence Amplitude ratio of two waves should be 3:2.
Reason:
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{{A_1} + {A_2}}}{{{A_1} - {A_2}}}} \right)^2}$
A.Both Assertion and reason are correct and Reason is the correct explanation for Assertion.
B.Both Assertion and Reason are correct but reason is not the correct explanation for Assertion.
C.Assertion is correct but reason is incorrect.
D.Assertion is incorrect but reason is correct.

Answer 1

Intensity of the wave is proportional to the square of the amplitude of the wave. Two waves when superimposed give maximum and minimum amplitude. Taking the ratio of maximum intensity and minimum intensity we can get a conclusion.
Formula:
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{{A_1} + {A_2}}}{{{A_1} - {A_2}}}} \right)^2}$

Complete answer:
Let us assume there are two waves of magnitude ${A_1}$and ${A_2}$, also suppose that ${A_1}$is greater in magnitude than ${A_2}$.When these two waves get superimposed, Amplitude of the resultant wave will have
Maximum Amplitude=${A_1} + {A_2}$
Minimum Amplitude= ${A_1} - {A_2}$
Now, we already know that intensity is directly proportional to the square of amplitude.
Therefore, for maximum and minimum intensity
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{{A_1} + {A_2}}}{{{A_1} - {A_2}}}} \right)^2}{\text{ }}..............{\text{(1)}}$
In the assertion it is given that
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{25}}{1}{\text{ }}...........{\text{(2)}}$
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{3}{2}{\text{ }}...........{\text{(3)}}$
Now, we can write equation (1) as
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{\dfrac{{{A_1}}}{{{A_2}}} + 1}}{{\dfrac{{{A_1}}}{{{A_2}}} - 1}}} \right)^2}$
Substitute eq. (3) in eq. (4)
We get,$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = 25$
So, the use in Assertion is given in the reason.
Hence, we can say that Assertion and reason both are correct and Reason is the correct explanation of reason.

Hence, Option(A) is correct.

Additional information:
We can define waves as disturbance travelling in a medium with certain speed. Due to vibration of particles in the medium, the disturbance that is the wave gets transferred from one particle to another. Wave motion is mainly of two types: Transverse motion and longitudinal motion. Wave Intensity can be defined as the power per unit area carried by a wave.
$I = \dfrac{P}{A}$.
Power is the rate at which energy is transferred by the wave through an Area A.

Note:
One must remember the formula of intensity and the concept of superposition to solve questions of wave motion. The intensity of the wave is directly proportional to the amplitude of the wave given. When amplitude increases, intensity increases. Intensity is proportional to the square of the amplitude.