Problem 14 - Master Resource Book in JEE Main Physics

Question

\(S_{1}\) and \(S_{2}\) are two coherent sources, The intensity of both sources are same. If the intensity at the point of maxima is \(4 \mathrm{Wm}^{-2}\), the intensity of each source is (a) \(1 \mathrm{Wm}^{-2}\) (b) \(2 \mathrm{Wm}^{-2}\) (c) \(3 \mathrm{Wm}^{-2}\) (d) \(4 \mathrm{Wm}^{-2}\)

Step-by-Step Solution

Answer

The intensity of each source is 1 Wm⁻².

Step 1: Understanding Interference and Intensity

In a double-slit experiment, when two coherent sources interfere constructively at a point of maxima, the resulting intensity is the sum of their individual intensities due to the formula for constructive interference: \( I_{ ext{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \).

Step 2: Given Information

We are given that \( I_1 = I_2 \) because the intensities of both sources are the same, and the intensity at the point of maxima is \( 4 \, \mathrm{Wm}^{-2} \).

Step 3: Setting Up the Equation

Since the sources are coherent and the intensities are equal, \( I_1 = I_2 = I \). The equation becomes: \( I_{ ext{max}} = (\sqrt{I} + \sqrt{I})^2 = (2\sqrt{I})^2 \). Substitute the given maxima intensity value: \( 4 = 4I \).

Step 4: Solving for Individual Intensity

Divide both sides of the equation by 4 to find \( I \): \( I = 1 \). Thus, the intensity of each source is \( 1 \, \mathrm{Wm}^{-2} \).

Key Concepts

Intensity Calculation

Intensity calculation is crucial in understanding wave interference, especially in a double-slit experiment. Intensity refers to the power of the wave per unit area and is typically measured in watts per square meter (W/m²). In the context of interference, particularly constructive interference, when two coherent sources, like the ones in the exercise, interfere at a maxima or bright spot, the intensity result is not just a simple addition. Since intensity is proportional to the square of the wave's amplitude, we use the relationship given by \[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \] when the intensities of the sources combine. This equation tells us that both the amplitude contributions and their phase coherence affect the final intensity level observed at the maxima.We are given that both sources have the same intensity, i.e., \( I_1 = I_2 = I \). Hence the equation simplifies further to \[ I_{\text{max}} = (2\sqrt{I})^2 = 4I \].By understanding how to manipulate and solve this equation, you can find the intensity of each individual source when given the maxima intensity.

Double-Slit Experiment

The double-slit experiment is fundamental in demonstrating the wave nature of light and other particles. This experiment involves two coherent light sources passing through two parallel slits, creating an interference pattern on a screen behind the slits. The key phenomenon here is the interference of the waves emanating from the slits, leading to alternating bright and dark bands called fringes. At points where the light waves from the slits combine constructively, bright fringes or points of maxima are observed. These bright spots occur due to the waves being in phase, reinforcing each other's amplitude and consequently increasing the intensity of the light at that point. This is the central concept when applied to calculations involving constructive interference. In practical application, such setups are used to measure wavelengths of light, the coherent property of sources, and delve deeper into the nature of wave interference. Understanding this experiment forms the basis for more complex studies in wave dynamics and quantum mechanics.

Constructive Interference

Constructive interference occurs when two waves meet and the resultant wave has a greater amplitude than either of the individual waves. This happens under the condition that the waves are in phase, meaning their peaks and troughs align perfectly.During a double-slit experiment, constructive interference is what gives rise to the bright spots on the interference pattern. Since the light waves from the two slits synchronously converge, they enhance each other and thus boost the intensity.The formula \[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \] arises from the amplitude relationship, where the total amplitude is the sum of individual amplitudes. If the sources are coherent and have the same intensity, then \( I_1 = I_2 = I \). Therefore, computing the maxima intensity simplifies greatly using this constructive interference principle.This concept is essential for analyzing not only light interference but can also be applied to sound waves, water waves, and even quantum particles, opening doors to a broader understanding of wave phenomena.