Answer to Question #214906 in Optics for lala

Question

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

Answer 1

$\text{condition of destructive interference:}$

$\Delta r = (2m-1)\frac{\lambda}{2}$

$\Delta r =|PA-PB|$

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

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

$PA +PB = AB;AB= 16$

$PB = AB - PA$

$|PA-PB| = |PA -(AB-PA)|= |2*PA -AB|=$

$|-(AB-2*PA)|= |AB-2*PA|= |16-2*PA|$

$\lambda =4cm$

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

$|16-2*PA|= 2*(2m-1);m\isin Z;$

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

$A.PA = 7$

$\text {condition(1) true}$

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

$B.PA = 8$

$\Delta r =|16-2*PA|=16-16=0$

$0 = 4m-2$

$m = \frac{1}{2};m\notin Z$

$\text {condition(1) false}$

$C.PA = 9$

$\text {condition(1) true}$

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

$D.PA = 11$

$\text {condition(1) true}$

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

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