Two coherent sources with intensity a interference..then ratio of I_max /Imin

When dealing with the interference of two coherent sources, understanding the relationship between intensity and the resulting interference pattern is crucial. The ratio of maximum intensity (I max) to minimum intensity (I min) can be derived from the principles of wave interference. Let’s break this down step by step.

Understanding Coherent Sources

Coherent sources are those that emit waves with a constant phase difference and the same frequency. This means that the waves produced by these sources will interact in a predictable manner, leading to constructive and destructive interference.

Intensity in Wave Interference

The intensity of a wave is proportional to the square of its amplitude. When two waves interfere, their amplitudes combine, leading to different intensity levels depending on whether the interference is constructive or destructive.

Constructive and Destructive Interference

• Constructive Interference: This occurs when the waves are in phase, meaning their peaks align. The resulting amplitude is the sum of the individual amplitudes, leading to maximum intensity.
• Destructive Interference: This happens when the waves are out of phase, meaning the peak of one wave aligns with the trough of another. The resulting amplitude can be reduced or even canceled out, leading to minimum intensity.

Calculating I max and I min

Let’s denote the amplitudes of the two coherent sources as A1 and A2. For simplicity, if we assume both sources have the same amplitude (A), the maximum and minimum intensities can be expressed as follows:

Maximum Intensity (I max)

When the waves interfere constructively, the total amplitude becomes:

A max = A1 + A2 = A + A = 2A

The intensity is then:

I max = (A max)² = (2A)² = 4A²

Minimum Intensity (I min)

In the case of destructive interference, the total amplitude is:

A min = A1 - A2 = A - A = 0

However, if the amplitudes are not equal, say A1 = A and A2 = kA (where k is a ratio of the amplitudes), the minimum intensity becomes:

A min = |A1 - A2| = |A - kA| = |(1 - k)A|

The intensity is then:

I min = (A min)² = [(1 - k)A]² = (1 - k)²A²

Finding the Intensity Ratio

Now, we can find the ratio of maximum intensity to minimum intensity:

Ratio = I max / I min = 4A² / (1 - k)²A²

After canceling out A², we have:

Ratio = 4 / (1 - k)²

Example Calculation

Let’s say both sources have equal amplitudes (k = 1). The ratio becomes:

Ratio = 4 / (1 - 1)²

This leads to an undefined situation since we cannot divide by zero. However, if k = 0.5, the ratio would be:

Ratio = 4 / (1 - 0.5)² = 4 / (0.5)² = 4 / 0.25 = 16

In summary, the ratio of maximum to minimum intensity in the interference pattern of two coherent sources is determined by their amplitude relationship. This ratio provides insight into the nature of the interference occurring between the waves.