(A)

The distance between two circular waves fronts is closest to,

[katex]\beta = \frac{{\lambda D}}{d}[/katex]

Here, [katex]\beta [/katex]

The distance between point P and the source 1 is proportional to the wavelength.

[katex]{r_{1P}} = 3\lambda [/katex]

Distance between point Q and point 1:

[katex]{r_{1Q}} = 3.5\lambda [/katex]

Distance between point R and point 1:

[katex]{r_{1R}} = 2.5\lambda [/katex]

(B)

The distance between two circular waves fronts is closest to,

[katex]\beta = \frac{{\lambda D}}{d}[/katex]

Here, [katex]\beta [/katex]

The distance between point P and the source 2 is proportional to the wavelength.

[katex]{r_{2P}} = 4\lambda [/katex]

Distance Q from source 2 is

[katex]{r_2}_Q = 2\lambda [/katex]

The distance between point R and the source 2 is also,

[katex]{r_{2R}} = 3.5\lambda [/katex]

(C)

If the path difference between wave fronts from points 1 and 2 equals an integer multiple of wavelength, the waves are considered constructively interfered. However, if it equals the odd multiple, the waves can be said to be destructively interfering.

The difference in the path between the waves that reach point P is called,

[katex]{\rm{Path difference = }}{r_{{\rm{2p}}}} – {r_{1p}}[/katex]

Here, [katex]{r_{{\rm{2p}}}}[/katex]

Substitute $equaTag12

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2p}}}} – {r_{1p}}} \right|\\\\ = 4\lambda – 3\lambda \\\\ = \left( 1 \right)\lambda \\\end{array}[/katex]

The path difference is an integer multiple of the wavelength so the interference is constructive.

(D)

The difference in the path between waves that reach Q at point Q is called,

[katex]{\rm{Path difference = }}{r_{{\rm{2Q}}}} – {r_{1Q}}[/katex]

Here, [katex]{r_{{\rm{2Q}}}}[/katex]

Substitute $equaTag16

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2Q}}}} – {r_{1Q}}} \right|\\\\ = \left| {3.5\lambda – 2\lambda } \right|\\\\ = \left( {1.5} \right)\lambda \\\end{array}[/katex]

The path difference equals half the wavelength of interference, so the interference is destructive.

(E)

The difference in the path between the waves that reach point R is

[katex]{\rm{Path difference = }}{r_{{\rm{2R}}}} – {r_{1R}}[/katex]

Here, [katex]{r_{{\rm{2R}}}}[/katex]

Substitute $equaTag20

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2R}}}} – {r_{1R}}} \right|\\\\ = \left| {2.5\lambda – 3.5\lambda } \right|\\\\ = \left( 1 \right)\lambda \\\end{array}[/katex]

The path difference equals an integer multiple of wavelength so the interference is constructive.

Part A – Ans

Distances from points P, Q and R to point 1 are [katex]3\lambda ,{\rm{ 3}}{\rm{.5}}\lambda {\rm{, and 2}}{\rm{.5}}\lambda [/katex]Part A

[katex]4\lambda ,{\rm{ 2}}\lambda {\rm{, 3}}{\rm{.5}}\lambda [/katex]Part B. The distance between points P, Q, and R are the following:

The interference at point P is constructive.

Part D

Interference at Q can be destructive.

Part E

The interference at point R is constructive.