Question

The ratio of intensities of rays emitted from two different coherent Sourees is \(\alpha .\) For the interference pattern by them, \(\left[\left(I_{\max }+I_{\min }\right) /\left(I_{\max }-I_{\min }\right)\right]\) will be equal to (A) \(\\{(1+\sqrt{\alpha}) / 2 \alpha\\}\) (B) \(\\{(1+\alpha) / 2 \alpha\\}\) (C) \(\\{(1+\sqrt{\alpha}) / 2\\}\) (D) \(\\{(1+\alpha) /(2 \sqrt{\alpha})\\}\)

Short answer

The short answer for the given question is: (D) \(\dfrac{(1+\alpha)}{(2 \sqrt{\alpha})}\)

Step-by-step solution

Find the relationship between I1 and I2 using given intensity ratio

Given that the ratio of intensities of rays emitted from two different coherent sources is: \(\alpha = \dfrac{I_1}{I_2}\) From the given ratio we can express \(I_1\) in terms of \(\alpha\), and \(I_2\): \(I_1 = \alpha I_2\) Next, we will find the intensities for the maximum and minimum interference patterns.

Find Imax and Imin

The maximum intensity \(I_{\max }\) occurs when the phase difference \(\theta = 0\). Hence, the formula for maximum interference pattern intensity is: \(I_{\max } = I_1 + I_2 + 2\sqrt{I_1 I_2}\) Similarly, the minimum intensity \(I_{\min }\) occurs when the phase difference \(\theta = \pi\). Hence, the formula for minimum interference pattern intensity is: \(I_{\min } = I_1 + I_2 - 2\sqrt{I_1 I_2}\)

Substitute I1 in terms of I2

Replace \(I_1\) with \(\alpha I_2\) in the equations for \(I_{\max }\) and \(I_{\min }\): \(I_{\max } = \alpha I_2 + I_2 + 2\sqrt{\alpha I_2^2}\) \(I_{\min } = \alpha I_2 + I_2 - 2\sqrt{\alpha I_2^2}\)

Calculate the expression for the given ratio

Now, we need to find the expression for \(\dfrac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }}\) Substitute the expressions obtained for \(I_{\max }\) and \(I_{\min }\) from the previous step: \(\dfrac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }} = \dfrac{(\alpha I_2 + I_2 + 2\sqrt{\alpha I_2^2}) + (\alpha I_2 + I_2 - 2\sqrt{\alpha I_2^2})}{(\alpha I_2 + I_2 + 2\sqrt{\alpha I_2^2}) - (\alpha I_2 + I_2 - 2\sqrt{\alpha I_2^2})}\) Simplify the expression: \(\dfrac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }} = \dfrac{2(\alpha I_2 + I_2)}{4\sqrt{\alpha I_2^2}}\) For this expression, notice how \(I_2\) cancels out in both numerator and denominator: \(\dfrac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }} = \dfrac{2(\alpha + 1)}{4\sqrt{\alpha}}\) Finally, simplify to get the required expression: \(\dfrac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }} = \dfrac{(1+\alpha)}{2\sqrt{\alpha}}\) Now, compare the expression with the given options to find the correct answer. The correct answer is: (D) \(\dfrac{(1+\alpha)}{(2 \sqrt{\alpha})}\)

Key concepts

Coherent Sources

Coherent sources are a fundamental concept when studying interference in physics. They are sources of waves that emit waves with a constant phase difference and the same frequency. This consistency is crucial for producing a stable interference pattern, which is when waves overlap and combine in predictable ways.
For instance, if you have two coherent light sources, the light waves they emit will align such that they can create patterns of constructive and destructive interference. Constructive interference occurs when the peak of one wave aligns with the peak of another, resulting in a brighter intensity. Conversely, destructive interference happens when the peak of one wave aligns with the trough of another, leading to a dimmer intensity or even complete cancellation.
Coherent sources can often be created using lasers. Lasers emit highly coherent light due to their design, where all light waves have the same frequency and phase. This coherence is what makes lasers suitable for experiments involving interference, leading to clear and distinct patterns.

Intensity in Interference

Intensity in interference is determined by how waves combine. It critically depends on the amplitudes of the interfering waves and the phase difference between them.
The intensity of an interference pattern can vary from maximum to minimum. When two waves are in phase (meaning their peaks and troughs align), they produce maximum intensity, which is given by the formula: \[ I_{\max} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]On the other hand, when the waves are out of phase (the peak of one wave aligns with the trough of another), they create minimum intensity:\[ I_{\min} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]The intensity depends on the square of the amplitude, so the combined intensity results from both the individual intensities and their interaction. A stronger interference pattern is created when the sources have equal intensities.

Phase Difference

Phase difference is a crucial element in understanding wave interference. It refers to the difference in phase between two interacting waves. This phase difference determines whether the interference is constructive or destructive.
In wave interference, a 0° phase difference leads to constructive interference, where the waves amplify each other. This results in higher intensity, as described by the equation: \[ I_{\max} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]Conversely, a 180° phase difference results in destructive interference, causing the waves to cancel each other out, thus leading to smaller intensity or total cancellation.In the context of the textbook exercise, the phase difference plays a pivotal role in calculating the ratio of intensities in an interference pattern. By understanding the phase difference, we can accurately predict the resulting wave pattern and intensity.