Introduction

When two coherent sources of light interfere, they produce an interference pattern characterized by bright and dark regions. The intensity of the light in these regions can vary, and it is possible to calculate the ratio of intensities of the two sources based on the variation in intensity.

Interference Pattern and Intensity Variation

In an interference pattern, the bright regions are caused by constructive interference, where the crests of the two waves align, resulting in increased intensity. The dark regions, on the other hand, are caused by destructive interference, where the crests of one wave align with the troughs of the other, resulting in decreased intensity.

The intensity variation in the interference pattern is defined as the difference between the maximum and minimum intensities divided by the average intensity:

Intensity Variation = (Imax - Imin) / Iavg

Given that the intensity variation is 5% of the average intensity, we can write:

0.05 = (Imax - Imin) / Iavg

Calculating the Ratio of Intensities

Let's assume the intensities of the two sources are I1 and I2, where I1 is the intensity of the first source and I2 is the intensity of the second source. We need to find the ratio of intensities, which can be expressed as:

Ratio = I1 / I2

To proceed further, we need to make some assumptions. Let's assume that the maximum intensity occurs when the waves from both sources are in phase, resulting in constructive interference. In this case, Imax can be expressed as:

Imax = I1 + I2

Similarly, let's assume that the minimum intensity occurs when the waves from both sources are completely out of phase, resulting in destructive interference. In this case, Imin can be expressed as:

Imin = I1 - I2

Substituting these values into the intensity variation equation, we have:

0.05 = (I1 + I2 - I1 + I2) / (I1 + I2)

Simplifying the equation, we get:

0.05 = 2I2 / (I1 + I2)

Now, we can rearrange the equation to solve for the ratio of intensities:

Ratio = I1 / I2 = (2 / 0.05) - 1

Ratio = 40 - 1 = 39

Conclusion

The ratio of intensities of the two coherent sources is 39. This means that the intensity of the first source is 39 times greater than the intensity of the second source. The intensity variation in the interference pattern can provide valuable information about the relative intensities of the sources involved in the interference phenomenon. Understanding this ratio can help in various applications, such as determining the power distribution between different sources or analyzing the interference patterns in optical systems.