In the adjacent diagram, CP represents a wavefront and AO & BP, the corresponding two rays. Find the condition on $θ$ for constructive interference at P between the ray BP and reflected ray OP. [2003]

Question

Smart Achievers · Accepted Answer

(2)

In $∆ O P M$ ,

$O P = \frac{d}{\cos θ}$

In $∆ C O P$ ,

$O C = \frac{d \cos 2 θ}{\cos θ}$

Path difference between the two rays reaching P

$= C O + O P + \frac{λ}{2} = \frac{d \cos 2 θ}{\cos θ} + \frac{d}{\cos θ} + \frac{λ}{2}$

$= \frac{d}{\cos θ} (\cos 2 θ + 1) + \frac{λ}{2} = 2 d \cos θ + \frac{λ}{2}$

For constructive interference at $P$ , path difference $= n λ$

$∴$ $2 d \cos θ + \frac{λ}{2} = n λ$ $\Rightarrow \cos θ = \frac{(2 n - 1) λ}{4 d}$

For $n = 1$ , $\cos θ = \frac{λ}{4 d}$

Solution

Q.
In the adjacent diagram, CP represents a wavefront and AO & BP, the corresponding two rays. Find the condition on $θ$ for constructive interference at P between the ray BP and reflected ray OP. [2003]

1 $\cos θ = \frac{3 λ}{2 d}$

2 $\cos θ = \frac{λ}{4 d}$

3 $\sec θ - \cos θ = \frac{λ}{d}$

4 $\sec θ - \cos θ = \frac{4 λ}{d}$

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Solution

Q. In the adjacent diagram, CP represents a wavefront and AO & BP, the corresponding two rays. Find the condition on θ for constructive interference at P between the ray BP and reflected ray OP. [2003]

1 cosθ=3λ2d

2 cosθ=λ4d

3 secθ-cosθ=λd

4 secθ-cosθ=4λd