The interference pattern is obtained with two coherent light sources of intensity ratio n. In the interference pattern, the ratio $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ will be

Question

Assertion (A): When two waves of equal amplitude and a phase difference of $ \frac{\pi}{2} $ interfere, the resulting intensity is equal to the intensity of one wave. Reason (R): In interference, the resultant intensity is always the sum of individual intensities if the phase difference is non-zero.

Answer 1

To solve this problem, we are dealing with the interference pattern created by two coherent light sources with an intensity ratio $ n $. We need to find the ratio $ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} $, where $ I_{max} $ and $ I_{min} $ represent the maximum and minimum intensities in the interference pattern, respectively.

First, let's recall that the maximum intensity $ I_{max} $ in an interference pattern is given by the constructive interference of two waves, and the formula can be expressed as: I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 where $ I_1 $ and $ I_2 $ are the intensities of the individual sources.
Similarly, the minimum intensity $ I_{min} $, due to destructive interference, is: I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2
The intensity ratio $ n $ is given by $ \frac{I_1}{I_2} $. Set $ I_1 = nI_2 $.
Substitute $ I_1 = nI_2 $ into the equations for $ I_{max} $ and $ I_{min} $:
- I_{max} = \left(\sqrt{nI_2} + \sqrt{I_2}\right)^2 = (\sqrt{n} \sqrt{I_2} + \sqrt{I_2})^2 = I_2 (\sqrt{n} + 1)^2
- I_{min} = \left(\sqrt{nI_2} - \sqrt{I_2}\right)^2 = (\sqrt{n} \sqrt{I_2} - \sqrt{I_2})^2 = I_2 (\sqrt{n} - 1)^2
Now, substitute these expressions into the required ratio: \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{I_2 (\sqrt{n} + 1)^2 - I_2 (\sqrt{n} - 1)^2}{I_2 (\sqrt{n} + 1)^2 + I_2 (\sqrt{n} - 1)^2}
Cancel $ I_2 $ from both numerator and denominator:
\frac{(\sqrt{n} + 1)^2 - (\sqrt{n} - 1)^2}{(\sqrt{n} + 1)^2 + (\sqrt{n} - 1)^2}
Simplify the numerator and the denominator:
- The numerator: \[ (\sqrt{n} + 1)^2 - (\sqrt{n} - 1)^2 = (n + 1 + 2\sqrt{n}) - (n + 1 - 2\sqrt{n}) = 4\sqrt{n} \]
- The denominator: \[ (\sqrt{n} + 1)^2 + (\sqrt{n} - 1)^2 = (n + 1 + 2\sqrt{n}) + (n + 1 - 2\sqrt{n}) = 2(n + 1) \]
Therefore, the required ratio is:
\frac{4\sqrt{n}}{2(n + 1)} = \frac{2\sqrt{n}}{n + 1}

Thus, the correct answer is \frac{2 \sqrt{n}}{n + 1}.

Answer 2

To solve this problem, we are dealing with the interference pattern created by two coherent light sources with an intensity ratio $ n $. We need to find the ratio $ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} $, where $ I_{max} $ and $ I_{min} $ represent the maximum and minimum intensities in the interference pattern, respectively.

First, let's recall that the maximum intensity $ I_{max} $ in an interference pattern is given by the constructive interference of two waves, and the formula can be expressed as: I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 where $ I_1 $ and $ I_2 $ are the intensities of the individual sources.
Similarly, the minimum intensity $ I_{min} $, due to destructive interference, is: I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2
The intensity ratio $ n $ is given by $ \frac{I_1}{I_2} $. Set $ I_1 = nI_2 $.
Substitute $ I_1 = nI_2 $ into the equations for $ I_{max} $ and $ I_{min} $:
- I_{max} = \left(\sqrt{nI_2} + \sqrt{I_2}\right)^2 = (\sqrt{n} \sqrt{I_2} + \sqrt{I_2})^2 = I_2 (\sqrt{n} + 1)^2
- I_{min} = \left(\sqrt{nI_2} - \sqrt{I_2}\right)^2 = (\sqrt{n} \sqrt{I_2} - \sqrt{I_2})^2 = I_2 (\sqrt{n} - 1)^2
Now, substitute these expressions into the required ratio: \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{I_2 (\sqrt{n} + 1)^2 - I_2 (\sqrt{n} - 1)^2}{I_2 (\sqrt{n} + 1)^2 + I_2 (\sqrt{n} - 1)^2}
Cancel $ I_2 $ from both numerator and denominator:
\frac{(\sqrt{n} + 1)^2 - (\sqrt{n} - 1)^2}{(\sqrt{n} + 1)^2 + (\sqrt{n} - 1)^2}
Simplify the numerator and the denominator:
- The numerator: \[ (\sqrt{n} + 1)^2 - (\sqrt{n} - 1)^2 = (n + 1 + 2\sqrt{n}) - (n + 1 - 2\sqrt{n}) = 4\sqrt{n} \]
- The denominator: \[ (\sqrt{n} + 1)^2 + (\sqrt{n} - 1)^2 = (n + 1 + 2\sqrt{n}) + (n + 1 - 2\sqrt{n}) = 2(n + 1) \]
Therefore, the required ratio is:
\frac{4\sqrt{n}}{2(n + 1)} = \frac{2\sqrt{n}}{n + 1}

Thus, the correct answer is \frac{2 \sqrt{n}}{n + 1}.

The interference pattern is obtained with two coherent light sources of intensity ratio n. In the interference pattern, the ratio $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ will be

The Correct Option is B

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