Two coherent sources produce waves of different intensities which interfere. After interference, the

Q: Why do we use square roots of intensities in the formula?

Because intensity is proportional to the square of the amplitude ($I \propto A^2$). When waves interfere, it is their amplitudes that add or subtract, not their intensities. So, $A_{max} = A_1 + A_2$, which translates to $\sqrt{I_{max}} = \sqrt{I_1} + \sqrt{I_2}$.

Q: What if the sources were incoherent?

If the sources were incoherent, they would not maintain a constant phase difference. The resultant intensity would just be the simple sum $I_1 + I_2$ everywhere on the screen, and no interference pattern (no distinct maxima or minima) would be formed.

Q: Can $I_{min}$ ever be zero?

Yes! If the two sources have equal intensities ($I_1 = I_2$), then $\sqrt{I_1} - \sqrt{I_2} = 0$, which means $I_{min} = 0$. This results in perfectly dark fringes, giving the best contrast.

Q: Why did we take the positive square root of 16?

Because $\sqrt{I_1}$ and $\sqrt{I_2}$ represent amplitudes, which are considered positive quantities in this context. The ratio of two positive quantities must be positive.

Q: Is this formula valid for any phase difference?

No, the specific formulas for $I_{max}$ and $I_{min}$ apply only to the points on the screen where the phase difference is an even multiple of $\pi$ (for maxima) and an odd multiple of $\pi$ (for minima). For any general phase difference $\phi$, the intensity is $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$.

Question

Two coherent sources produce waves of different intensities which interfere. After interference, the ratio of the maximum intensity to the minimum intensity is 16. The intensity of the waves are in the ratio

Options:

(A) 16 : 9

(B) 5 : 3

(C) 25 : 9

(D) 4 : 1

Sigma Percentile · Accepted Answer

### The Dance of Light: Decoding Interference Intensities

Imagine you are looking at a screen where light from two coherent sources is overlapping. These aren't just any light sources; because they are coherent, their waves maintain a constant phase relationship, allowing them to perform a beautiful dance of constructive and destructive interference. Let's call the individual intensities of these sources $I_1$ and $I_2$.

We are given a very specific clue about the interference pattern they create: the ratio of the brightest spot on the screen to the darkest spot is exactly $16$.

### The Mathematics of Maxima and Minima

To unlock this puzzle, we need to recall the fundamental formulas for the maximum and minimum intensities in an interference pattern. The maximum intensity occurs when waves add up constructively (crest meets crest), and the minimum occurs when they cancel out destructively (crest meets trough).

The formulas are:
$$I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2$$
$$I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2$$

The problem states that the ratio of maximum to minimum intensity is $16$. Let's substitute our formulas into this ratio. This gives us a nice algebraic equation to work with:

$$\frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} = 16$$

### Solving the Puzzle: From Ratio to Amplitudes

To simplify this bulky expression, let's take the square root on both sides. The square root of $16$ is $4$. This removes the squares and makes the algebra much easier.

$$\frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = 4$$

Now, we just cross-multiply and group the like terms.

$$\sqrt{I_1} + \sqrt{I_2} = 4\sqrt{I_1} - 4\sqrt{I_2}$$

Bring all the $I_1$ terms to one side and the $I_2$ terms to the other:

$$5\sqrt{I_2} = 3\sqrt{I_1}$$

We find that the ratio of the square roots of the intensities, which is essentially the ratio of their wave amplitudes, is:

$$\frac{\sqrt{I_1}}{\sqrt{I_2}} = \frac{5}{3}$$

### The Final Reveal: Intensity Ratio

Finally, to get the ratio of the original intensities, we just need to square this result. Squaring $5/3$ gives us our final answer.

$$\frac{I_1}{I_2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$$

**The intensities of the waves are in the ratio 25 : 9.**

As a thought experiment, what if the sources were incoherent instead of coherent? In that case, the intensities would simply add up everywhere ($I_{resultant} = I_1 + I_2$), and we wouldn't see any interference pattern at all. Keep this distinction in mind for future problems!

Animated Solution for Physics - Optics: Two coherent sources produce waves of different intensities which interfere. After interference, the ratio of the maximum intensity to the minimum intensity is 16. The intensity of the waves are in the ratio

Select Answer:

Visualized Solution

Initial Setup

Intensity Extrema

Applying the Condition

Simplifying the Equation

Solving for Amplitudes

Final Intensity Ratio

Food for Thought

The Sigma Insight: Interference and Young's Double-Slit Experiment

The Dance of Light

The Mathematics of Maxima and Minima

Solving the Puzzle

The Final Reveal

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