Animated Solution for Physics - Optics: Two coherent monochromatic light beams of intensities $I$ and $4I$ are superposed. The maximum and minimum possible intensities in the resulting beam are

Have you ever watched two water ripples collide and create a larger wave, or sometimes cancel each other out completely? Light does the exact same thing! This beautiful phenomenon is called interference, and it's all about how waves add up.

But what we actually see is the intensity of the light, which is proportional to the square of the amplitude.

If two waves have intensities $I_1$ and $I_2$, and a phase difference $\varphi$, the resultant intensity is given by the master equation:

The term $2\sqrt{I_1{I}_2}\cos \varphi$ is the magic ingredient—the interference term. It dictates whether the waves will work together or fight against each other.

The Maximum Intensity When the waves are perfectly in sync (in phase), the crests align with crests. Mathematically, this happens when $\varphi =0,2\pi ,4\pi \dots$, making $\cos \varphi =1$.

The intensity reaches its absolute peak:

In our problem, we are given $I_1=4I$ and $I_2=I$. Let's plug these in:

Notice how the maximum intensity ($9I$) is significantly larger than just the simple sum of the two intensities ($5I$). The waves are amplifying each other!

The Minimum Intensity Now, imagine the waves are completely out of sync (out of phase). A crest meets a trough. This occurs when $\varphi =\pi ,3\pi ,5\pi \dots$, making $\cos \varphi =-1$.

The intensity drops to its lowest point:

Substituting our values again:

Even though we are shining two lights together, the resulting minimum intensity is just $I$. The waves have partially cancelled each other out.

So, the intensity of the resulting beam will fluctuate between a dazzling $9I$ and a dim $I$, depending on the phase difference at any given point in space.

A quick thought experiment: What if the two light sources were incoherent (like two separate light bulbs)? Incoherent sources have a rapidly and randomly changing phase difference. The average of $\cos \varphi$ over time is zero. The interference term vanishes, and the total intensity is just the simple sum everywhere: $I_{avg}=4I+I=5I$. No bright and dark fringes, just a uniform glow!