Question

If the sources A and B in the figure (Figure 1) are emitting waves of wavelength λ that are in phase with each other, constructive interference will occur at point P if (there may be more than one correct choice):

A. x=y.
B. x+y=λ.
C. x−y=2λ.
D. x−y=5λ.

Answer Options:

A, B, CD, ACD, ABC, BD, AC

Answer 1

Concepts and reason

The concepts used in this problem are constructive interference and path difference.

First, explain the constructive interference by the diagram and choose the wrong option for path difference for two wavelengths.

Later on, choose the correct option for the path difference of emitting waves by using the concept of path difference.

Fundamentals

Constructive interference:

Constructive interference has in-phase waves. In-phase waves are those waves in which peak point of the waves are located at the same position in the wave cycle. These are separated by an even multiple of the half wavelength. The phase shift between these waves is $0^\circ$ .

Path difference:

Path difference causes due to two waves. It is measured in wavelength. Path difference has a direct relationship with a phase difference.

The expression for path difference is given by,

$\Delta d = n\lambda$

Here, $\Delta d$ is path difference, $n$ is the integer value and $\lambda$ is the wavelength.

The value of $n$ may be $0,{\rm{ }} \pm 1,{\rm{ }} \pm 2,{\rm{ }} \pm 3....$ .

Consider that A and B are two coherent sources and emitting two waves of wavelength $\lambda$ . These two waves have their peak points at the same position and the phase shift between these waves is $0^\circ$ .

The diagram shows the constructive interference of the two waves.

Refer to the above figure; the length of the wave from source A is shorter than the length of the wave from source B. The phase angle between the peak points of those waves is $0^\circ$ .

The general expression for path difference is given by,

$\Delta d = n\lambda$

But the path difference for the above diagram is $x - y$ .

Substitute $x - y$ for $\Delta d$ in the above equation.

$x - y = n\lambda$ ...... (1)

Here, $x$ is the length of the wave from source B and $y$ is the length of the wave from source A.

Substitute 1 for $n$ in the equation (1).

$x - y = \lambda$

From the above equation, for $n = 1$ , the path difference for two waves is $x - y = \lambda$ . This does not follow the option.

Hence, for $n = 1$ , $x + y = \lambda$ is incorrect.

The expression for the path difference for the question is given by,

$x - y = n\lambda$ ...... (2)

Substitute 0 for $n$ in the equation (2).

$\begin{array}{c}\\x - y = \left( 0 \right)\lambda \\\\x - y = 0\\\\x = y\\\end{array}$

Hence, option (A) is correct.

Substitute 2 for $n$ in the equation (2).

$x - y = 2\lambda$

Hence, option (C) is correct.

Substitute 5 for $n$ in the equation (2).

$x - y = 5\lambda$

Hence, the option (D) is correct.

Ans:

The required path differences for the wave are:

$\begin{array}{c}\\x = y\\\\x - y = 2\lambda \\\\x - y = 5\lambda \\\end{array}$