Two waves of intensity ratio $1: 9$ cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is $\mathrm{I}_{1}$ (b) Waves are coherent is $I_{2}$ and differ in phase by $60^{\circ}$. If $\frac{I_{1}}{I_{2}}=\frac{10}{x}$ then $\mathrm{x}=$ ____.

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Wave Optics

Two waves of intensity ratio

1 : 9

cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is

I_{1}

(b) Waves are coherent is

I_{2}

and differ in phase by

6 0^{\circ}

. If

\frac{I _{1}}{I _{2}} = \frac{10}{x}

then

x =

____.

Answer: 13

Solution

Let the individual intensities be

I_{0}

and

9 I_{0}

.

For incoherent waves, intensities simply add:

I_{1} = I_{0} + 9 I_{0} = 10 I_{0}

For coherent waves differing in phase by

6 0^{\circ}

I_{2} = I_{0} + 9 I_{0} + 2 I_{0} \cdot 9 I_{0} cos 6 0^{\circ} = 10 I_{0} + 2 (3 I_{0}) (\frac{1}{2}) = 13 I_{0}

\frac{I _{1}}{I _{2}} = \frac{10}{13} = \frac{10}{x} ⟹ x = 13

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Integer answer

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