Animated Solution for Physics - Optics: Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $\frac{I_{\text{max}}-I_{\text{min}}}{I_{\text{max}}+I_{\text{min}}}$ will be

Have you ever watched the mesmerizing patterns of colors on a soap bubble or the intricate ripples when two stones are dropped in a pond? These beautiful phenomena are governed by the principle of superposition, which leads to interference. In this problem, we are diving deep into the mathematics of the interference pattern produced by two coherent light sources.

When two coherent light waves meet, they don't just add up like simple numbers. Because light is a wave, it has both an amplitude and a phase. If the crest of one wave meets the crest of another, they build each other up, creating a bright spot. This is called constructive interference. Conversely, if a crest meets a trough, they cancel each other out, creating a dark spot. This is known as destructive interference.

The intensity of light is directly proportional to the square of its amplitude. When we mathematically combine two waves with intensities $I_1$ and $I_2$, the resultant intensity $I$ at any point on the screen depends on the phase difference $\varphi$ between them:

Maximum and Minimum Intensity

To find the brightest and darkest points on the screen, we look at the extremes of the cosine function. The maximum intensity occurs when $\cos \varphi =1$ (constructive interference):

The minimum intensity occurs when $\cos \varphi =-1$ (destructive interference):

These two master equations are the keys to unlocking our problem.

The Intensity Ratio

The problem states that the ratio of the intensities of the two coherent sources is $2x$. Mathematically, we can write this as:

This allows us to express the intensity of the first source in terms of the second:

Simplifying the Visibility Expression

We are asked to find the value of a specific ratio: $\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}$. Before we rush to substitute our $x$ values, let's simplify this expression algebraically. It will save us a lot of messy calculations!

First, let's find the numerator by subtracting $I_{\min }$ from $I_{\max }$:

Next, let's find the denominator by adding them together:

Now, we divide the numerator by the denominator to get our simplified ratio:

Substituting and Solving

Now that we have a beautifully simplified expression, we can substitute our relationship $I_1=2x{I}_2$ into it:

Let's clean up the numerator and denominator by factoring out $I_2$:

Since $I_2$ is positive, we can pull it out of the square root in the numerator:

Finally, the $I_2$ terms cancel out perfectly, leaving us with our final answer:

The Physical Significance of Fringe Visibility

You might be wondering, does this specific ratio have a physical meaning? Yes, it does! The quantity $\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}$ is known in optics as the Fringe Visibility or Contrast.

It tells us how sharp and distinct the interference fringes are. If the two sources have equal intensity ($I_1=I_2$), the minimum intensity drops to zero, and the visibility becomes exactly $1$ (or $100\%$). As the intensities become more unequal, the dark fringes aren't completely dark anymore, and the visibility drops, making the pattern look washed out. Understanding this mathematical ratio gives you a direct window into the physical reality of the experiment!