Question

What is the maximum intensity of interference of $n$ identical waves each of intensity $I_0$, if the sources are coherent and incoherent respectively.

• A. $n^2{I}_0,n^2{I}_0$
• B. $n{I}_0,n{I}_0$
• C. $n{I}_0,n^2{I}_0$
• D. $n^2{I}_0,n{I}_0$

Answer 1

The correct option is D $n^2{I}_0,n{I}_0$

We know that, for two waves, the resultant intensity is given by,

$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$

The sources are said to be coherent if they have constant phase difference $\left(\varphi \right)$ between them. The intensity will be maximum, when $\varphi =2n\pi ;$ the sources are in the same phase.

Thus,
$I_{max}=I_1+I_1+2\sqrt{I_1{I}_2}=(\sqrt{I_1}+\sqrt{I_2}{)}^2$

Similarly, for $n$ identical waves,

$I_{max}=(\sqrt{I_0}+\sqrt{I_0}+\sqrt{I_0}+.....{)}^2=n^2{I}_0$

Further, the incoherent sources have phase difference that varies randomly with time.

Thus, taking average over one cycle, $<\cos \varphi >=0$

Hence, $I=I_1+I_2$

Similarly, for n identical waves,

$I=I_0+I_0+I_0+........=n{I}_0$

Therefore, option $\left(\text{D}\right)$ is correct.