In Young's double-slit experiment, the two slits act as coherent sources of equal amplitude A and of

Q: Why is the phase difference zero at the mid-point?

Because the mid-point is equidistant from both slits. The waves travel the exact same distance, so their path difference is zero. A zero path difference directly translates to a zero phase difference.

Q: What exactly does 'incoherent' mean?

Incoherent means the two sources do not maintain a constant phase relationship. They emit light waves randomly, so the phase difference between them is constantly changing in a chaotic manner.

Q: Why do we just add intensities for incoherent sources?

Because the phase difference changes so rapidly, our eyes (or any standard detector) can only measure the time-averaged intensity. The average of $\cos\phi$ over time is zero, so the interference term $2\sqrt{I_1 I_2}\cos\phi$ vanishes, leaving just $I_1 + I_2$.

Q: Can two separate light bulbs act as coherent sources?

No. Independent light sources emit light through spontaneous emission from billions of different atoms randomly. It's impossible for them to maintain a constant phase difference. That's why YDSE uses a single source split into two.

Q: How is intensity related to amplitude?

Intensity is directly proportional to the square of the amplitude ($I \propto A^2$). If two sources have the same amplitude $A$, they will have the same individual intensity $I_0$.

Question

In Young's double-slit experiment, the two slits act as coherent sources of equal amplitude $A$ and of wavelength $\lambda$. In another experiment with the same set-up the two slits are sources of equal amplitude $A$ and wavelength $\lambda$, but are incoherent. The ratio of the intensity of light at the mid-point of the screen in the first case to that in the second case is ......

Sigma Percentile · Accepted Answer

The magic of Young's Double Slit Experiment (YDSE) lies in its profound demonstration of the wave nature of light. But what happens when we tweak the rules of the game? What if the sources of light stop cooperating and start acting independently? This question takes us on a fascinating journey through the concepts of coherence, incoherence, and the mathematics of wave interference.

### The Nature of Light and Intensity

Before we dive into the experiment, let's establish a fundamental rule. Light is an electromagnetic wave, and like all waves, it has an amplitude, which we will call $A$. The brightness, or **intensity ($I$)**, of this light is directly proportional to the square of its amplitude.

$$I \propto A^2$$

In our problem, both slits act as sources emitting light with the same amplitude $A$. Therefore, if we were to measure the intensity of the light coming from just one slit, it would be a constant value, let's call it $I_0$.

### Case 1: The Symphony of Coherent Sources

In the classic YDSE, the two slits are illuminated by a single primary source. This ensures that the light waves emerging from the two slits are **coherent**. Coherence means that the waves maintain a constant phase relationship with each other. They are like two perfectly synchronized dancers.

When these coherent waves meet on the screen, they interfere. The master equation for the resultant intensity $I$ of two interfering waves is:

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

Here, $\phi$ is the phase difference between the waves at the point they meet.

Now, let's focus on the **mid-point** of the screen. Because this point is exactly equidistant from both slits, the waves travel the exact same distance to get there. The path difference is zero, which means the phase difference $\phi$ is also exactly zero ($0^\circ$).

Substituting $\phi = 0$ and $I_1 = I_2 = I_0$ into our master equation:

$$I_{coh} = I_0 + I_0 + 2\sqrt{I_0 \cdot I_0}\cos(0^\circ)$$

Since $\cos(0^\circ) = 1$, the equation simplifies beautifully:

$$I_{coh} = 2I_0 + 2I_0 = 4I_0$$

This is the phenomenon of **constructive interference**. The waves arrive perfectly in step, crest to crest and trough to trough, creating a bright spot that is four times as intense as a single slit!

### Case 2: The Chaos of Incoherent Sources

Now, imagine we replace our setup with two completely independent light sources, like two separate tiny light bulbs. These sources are **incoherent**. They emit light waves in random bursts, meaning the phase difference $\phi$ between them is constantly and unpredictably changing.

Because the phase difference is fluctuating millions of times per second, our eyes (or any standard light detector) cannot keep up. Instead, we observe the **time-averaged** intensity.

Let's look back at our master equation:

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

When we take the average over time, the values of $I_1$ and $I_2$ remain constant. However, the $\cos\phi$ term is wildly swinging between $+1$ and $-1$. The average value of a cosine function over many random cycles is exactly zero!

$$\langle \cos\phi angle = 0$$

Because of this, the entire interference term ($2\sqrt{I_1 I_2}\cos\phi$) vanishes into thin air. We are left with a simple addition of intensities:

$$I_{incoh} = I_1 + I_2 = I_0 + I_0 = 2I_0$$

Without coherence, there is no interference pattern. The light simply adds up, giving us an intensity of $2I_0$ everywhere on the screen, including the mid-point.

### The Grand Finale: Calculating the Ratio

The problem asks for the ratio of the intensity at the mid-point in the first case (coherent) to that in the second case (incoherent).

$$	ext{Ratio} = \frac{I_{coh}}{I_{incoh}}$$

Substituting the values we derived:

$$	ext{Ratio} = \frac{4I_0}{2I_0} = 2$$

The intensity at the center of the interference pattern is exactly twice as bright as it would be if the sources were just randomly shining light together. This elegant result highlights the incredible power of wave interference!

Enter Numerical Value:

Visualized Solution

Intensity of a Single Source

Case 1: Coherent Sources

Case 2: Incoherent Sources

Ratio of Intensities

The Sigma Insight: Interference of Waves

The Nature of Light and Intensity

Case 1

Case 2

The Grand Finale

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