Question

Which of the following is the path difference for destructive interference?
$\text{A.}n(\lambda +1)$
$\text{B.}(2n+1){\displaystyle \frac{\lambda }{2}}$
$\text{C.}n\lambda$
$\text{D.}(n+1){\displaystyle \frac{\lambda }{2}}$

Answer 1

Hint: Destructive interference at a point means at that point, the intensity of light wave or simply wave is the minimum. This point is also called minima. Constructive interference at a point means at that point, the intensity of light wave or simply wave is the maximum. This point is also called maxima.

Formula Used:
$Phasedifference(\varphi )={\displaystyle \frac{2\pi }{\lambda }}\times Pathdifference(\mathrm{\Delta }x)$

Complete step-by-step answer:
The amount of intensity present at a point is the function of the phase difference at that point. If the phase difference between two waves is $\pi ,3\pi ,5\pi ...$ i.e. the two waves are out of phase, then the resulting intensity due to two waves is minimum.
Hence we can say, for destructive interference,$\varphi =(2n+1)\pi$, here $\varphi$ is called phase difference.
Now, as we know the relation between phase difference and path difference;
$Phasedifference={\displaystyle \frac{2\pi }{\lambda }}\times Pathdifference$
As $\varphi =(2n+1)\pi$, putting in the equation, we get;
$(2n+1)\pi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x$
Or $\mathrm{\Delta }x=(2n+1){\displaystyle \frac{\lambda }{2}}$
Hence, we can say that for destructive interference, the path difference must be an odd multiple of ${\displaystyle \frac{\lambda }{2}}$, i.e.$\mathrm{\Delta }x=(2n+1){\displaystyle \frac{\lambda }{2}},nϵ[0,1,2,3...)$
Diagram showing the condition when two waves undergo destructive interference.

Here we can see that all the points on one wave have a phase difference of $\pi$ with respect to the corresponding points on the second wave.

Note: In the above diagram, we can see that the destructive interference occurs if every point on one wave has a phase difference of $\pi$ or odd multiples of $\pi$. The logic behind this is that the point on a wave represents the motion or displacement of a particle. If both waves get superimposed, then the net displacement of the point results by adding the displacements of corresponding points on the waves. Similarly for constructive interference, corresponding points on the two waves add in such a way that the net displacement of particles after superimposition of waves is maximum.