Problem 61 - Master Resource Book in JEE Main Physics

Question

In Young's double slit experiment, distance between two sources is \(0.1 \mathrm{~mm}\). The distance of screen from the sources is \(20 \mathrm{~cm}\). Wavelength of light used is \(5460 \AA\). Then angular position of first dark fringe is (a) \(0.08^{\prime}\) (b) \(0.16^{\circ}\) (c) \(0.20^{\circ}\) (d) \(032^{\circ}\)

Step-by-Step Solution

Answer

The angular position of the first dark fringe is \(0.16^{\circ}\), which corresponds to option (b).

Step 1: Convert Units

First, convert the given units to a consistent system. The parameters given are in millimeters and centimeters, which we need to convert to meters for standard SI units. The distance between the sources, \(d\), is \(0.1 \text{ mm} = 0.1 \times 10^{-3} \text{ m}\). The distance to the screen, \(D\), is \(20 \text{ cm} = 0.2 \text{ m}\). The wavelength of light, \(\lambda\), is \(5460 \text{\AA} = 5460 \times 10^{-10} \text{ m}\).

Step 2: Understand the Condition for the First Dark Fringe

In Young's double-slit experiment, the first dark fringe occurs when the path difference is half the wavelength, i.e., \((2m+1)\frac{\lambda}{2}\). For the first dark fringe (\(m=0\)), the path difference is \(\frac{\lambda}{2}\). The formula for the angular position \(\theta\) of a dark fringe is given by: \(d \sin \theta = (m+\frac{1}{2}) \lambda\).

Step 3: Calculate the Angular Position

Since we are looking for the first dark fringe, substitute \(m=0\) into the formula: \(d \sin \theta = \frac{\lambda}{2}\). Substituting the known values: \(0.1 \times 10^{-3} \sin \theta = \frac{5460 \times 10^{-10}}{2}\). Solve for \(\sin \theta\): \(\sin \theta = \frac{5460 \times 10^{-10}}{2 \times 0.1 \times 10^{-3}}\).

Step 4: Approximate \(\theta\) for Small Angles

Since the angle is small, \(\sin \theta \approx \theta\) in radians. Calculate \(\theta\): \(\theta = \frac{5460 \times 10^{-10}}{0.2 \times 10^{-3}}\).

Step 5: Convert Radians to Degrees

To convert \(\theta\) from radians to degrees, use the conversion \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\). Calculate: \(\theta \text{ in degrees} = \left(\frac{5460 \times 10^{-10}}{0.2 \times 10^{-3}}\right) \times \frac{180}{\pi}\).

Step 6: Choose the Correct Answer

Calculate and compare the result to the provided options: \(\theta \approx 0.16^{\circ}\). Therefore, the correct answer is option \((b)\).

Key Concepts

Understanding Angular Position in Young's Double Slit Experiment

In Young's double slit experiment, the concept of angular position is pivotal. This angle, which we denote as \(\theta\), helps determine the location of bright and dark fringes on the screen. In simple terms, angular position refers to the angle at which a particular fringe appears relative to the central axis of the setup.

This is crucial because the pattern of light and dark bands you observe is a result of light waves interacting constructively and destructively at these specific angles. To find the angular position \(\theta\) of a dark fringe, we typically use the formula \(d \sin \theta = (m+\frac{1}{2}) \lambda\) where:

• \(d\) is the distance between the slits (in meters).
• \(m\) is the order number of the fringe (\(m=0\) for the first dark fringe).
• \(\lambda\) is the wavelength of light (in meters).

Since \(\sin \theta\) often approximates \(\theta\) for small values, this relationship allows us to approximate \(\theta\) directly in radians. This small-angle approximation simplifies the mathematics involved, making it more accessible when analyzing fringe patterns.

Finding the First Dark Fringe in the Pattern

In the light pattern created by Young's double slit experiment, the first dark fringe is of particular interest. This fringe emerges when the path difference between waves from adjacent slits is such that they destructively interfere. A path difference of \(\frac{\lambda}{2}\) leads to this scenario.

We focus on the condition for these dark fringes through the formula \(d \sin \theta = (m+\frac{1}{2}) \lambda\). For the first dark fringe, \(m=0\), simplifying the equation to \(d \sin \theta = \frac{\lambda}{2}\). This equation gives us a direct path to calculate the angular position of the first dark fringe from the setup parameters.

Understanding this condition is crucial in experimental situations, as it helps to pinpoint where these dark bands will appear and thus verify theoretical predictions. It is a fascinating phenomenon where science meets real-world application, allowing us to see wave interference in action.

The Conversion of Wavelengths for Accurate Calculations

Understanding measurements in standard units is a key part in solving physics problems accurately, and wavelength conversion is essential in Young's double slit experiment. In this experiment, light wavelengths are often provided in Angstroms (\(\text{\AA}\)) but must be converted to meters (\(\text{m}\)) because the SI unit system is universally used in physics calculations.

The conversion is simple: \(1 \text{\AA} = 10^{-10} \text{ m}\). Given a wavelength, \(5460 \text{\AA}\), it converts to \(5460 \times 10^{-10} \text{ m}\).

This conversion is vital because it standardizes all measurements, ensuring calculations remain consistent and accurate. Without properly converting units, results can become misleading. This is particularly important in calculating the precise angular position of fringes where precision is necessary to validate theoretical models and experiments.