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Physics-

General

Easy

Question

In the adjacent diagram, $C P$ represents a wavefront and $A O$ and $B P$ , the corresponding two rays. Find the condition on $theta$ for constructive interference at P between the ray $B P$ and reflected ray $O P$

$\cos θ = \frac{3 λ}{2 d}$
$\cos θ = \frac{λ}{4 d}$
$\sec θ - \cos θ = \frac{λ}{d}$
$\sec θ - \cos θ = \frac{4 λ}{d}$

The correct answer is: $cos invisible function application theta equals fraction numerator lambda over denominator 4 d end fraction$

$P R equals d$
$P O equals d sec invisible function application theta$
And $C O equals P O cot invisible function application blank 2 theta equals d sec invisible function application theta cos invisible function application 2 theta$
Path difference between the two rays is,
$increment x equals C O plus P O$
$equals left parenthesis d sec invisible function application theta plus blank d sec invisible function application theta cos invisible function application 2 theta right parenthesis$
Phase difference between the two rays is
$increment ϕ equals pi$ (one is reflected, while another is direct)
Therefore condition for constructive interference should be

$capital delta x equals fraction numerator lambda over denominator 2 end fraction comma fraction numerator 3 lambda over denominator 2 end fraction horizontal ellipsis.$
Or $d sec invisible function application theta open parentheses 1 plus cos invisible function application 2 theta close parentheses equals fraction numerator lambda over denominator 2 end fraction$
Or $open parentheses fraction numerator d over denominator cos invisible function application theta end fraction close parentheses open parentheses 2 cos to the power of 2 end exponent invisible function application theta close parentheses equals fraction numerator lambda over denominator 2 end fraction$
Or $cos invisible function application theta equals fraction numerator lambda over denominator 4 d end fraction$

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