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When two coherent monochromatic light beams of intensities I and 4I are superimposed, the ratio between maximum and minimum intensities in the resultant beam is

  • a)
    9:1

  • b)
    1:9

  • c)
    4:1

  • d)
    1:4

Correct answer is option 'A'. Can you explain this answer?
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Answer:

Concept: When two coherent monochromatic light beams of intensities I and 4I are superimposed, the ratio between maximum and minimum intensities in the resultant beam is given by the formula:

$\frac{I_{max}}{I_{min}} = \frac{(I_1 + I_2) + \sqrt{(I_1 + I_2)^2 - 4I_1I_2}\cos\phi}{(I_1 + I_2) - \sqrt{(I_1 + I_2)^2 - 4I_1I_2}\cos\phi}$

Where I1 and I2 are the intensities of the two coherent monochromatic light beams, and φ is the phase difference between them.

Calculation:

Given, intensities of the two coherent monochromatic light beams are I and 4I.

So, I1 = I and I2 = 4I

Let's substitute these values in the above formula:

$\frac{I_{max}}{I_{min}} = \frac{(I_1 + I_2) + \sqrt{(I_1 + I_2)^2 - 4I_1I_2}\cos\phi}{(I_1 + I_2) - \sqrt{(I_1 + I_2)^2 - 4I_1I_2}\cos\phi}$

$\frac{I_{max}}{I_{min}} = \frac{(I + 4I) + \sqrt{(I + 4I)^2 - 4I(4I)}\cos\phi}{(I + 4I) - \sqrt{(I + 4I)^2 - 4I(4I)}\cos\phi}$

$\frac{I_{max}}{I_{min}} = \frac{5I + \sqrt{25I^2 - 16I^2}\cos\phi}{5I - \sqrt{25I^2 - 16I^2}\cos\phi}$

$\frac{I_{max}}{I_{min}} = \frac{5I + 3I\cos\phi}{5I - 3I\cos\phi}$

To find maximum and minimum intensities, we need to find the maximum and minimum values of cosφ.

When cosφ = 1 (maximum value), we get:

$\frac{I_{max}}{I_{min}} = \frac{5I + 3I}{5I - 3I} = \frac{8I}{2I} = 4$

When cosφ = -1 (minimum value), we get:

$\frac{I_{max}}{I_{min}} = \frac{5I - 3I}{5I + 3I} = \frac{2I}{8I} = \frac{1}{4}$

Therefore, the ratio between maximum and minimum intensities in the resultant beam is 4:1, which is option A.

Therefore, the correct answer is option A.
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When two coherent monochromatic light beams of intensities I and 4I are superimposed, the ratio between maximum and minimum intensities in the resultant beam isa)9:1b)1:9c)4:1d)1:4Correct answer is option 'A'. Can you explain this answer?
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