Question #214906

Coherent sources A and B emit electromagnetic waves with a wavelength of 4 cm. The distance between A and B is 16.0 cm. Point P is located somewhere between A and B. Which of the following values is/are the possible distance of P from A, if it is a point of destructive interference?

A. 7 cm

B. 8 cm

C. 9 cm

D. 11 cm


1
Expert's answer
2021-07-16T08:46:16-0400

condition of destructive interference:\text{condition of destructive interference:}

Δr=(2m1)λ2\Delta r = (2m-1)\frac{\lambda}{2}

Δr=PAPB\Delta r =|PA-PB|

By the condition of the problem\text{By the condition of the problem}

Point P is located somewhere between A and B:\text{Point P is located somewhere between A and B:}

PA+PB=AB;AB=16PA +PB = AB;AB= 16

PB=ABPAPB = AB - PA

PAPB=PA(ABPA)=2PAAB=|PA-PB| = |PA -(AB-PA)|= |2*PA -AB|=

(AB2PA)=AB2PA=162PA|-(AB-2*PA)|= |AB-2*PA|= |16-2*PA|

λ=4cm\lambda =4cm

then condition of destructive interference:\text{then condition of destructive interference:}

162PA=2(2m1);mZ;|16-2*PA|= 2*(2m-1);m\isin Z;

162PA=4m2;mZ;(1)|16-2*PA|= 4m-2;m\isin Z;(1)


A.PA=7A.PA = 7

condition(1) true\text {condition(1) true}

1614=4m2 for m=1|16-14|=4m-2 \text{ for }m=1

B.PA=8B.PA = 8

Δr=162PA=1616=0\Delta r =|16-2*PA|=16-16=0

0=4m20 = 4m-2

m=12;mZm = \frac{1}{2};m\notin Z

condition(1) false\text {condition(1) false}

C.PA=9C.PA = 9

condition(1) true\text {condition(1) true}

1618=4m2 for m=1|16-18|=4m-2 \text{ for }m=1

D.PA=11D.PA = 11

condition(1) true\text {condition(1) true}

1622=4m2 for m=2|16-22|=4m-2 \text{ for }m=2


Answer:A,C,D\text{Answer:} A,C,D


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