Problem 35 - Master Resource Book in JEE Main Physics

Question

In a biprism experiment, by using light of wavelength \(5000 \AA, 5 \mathrm{~mm}\) wide fringes are obtained on a screen \(1.0 \mathrm{~m}\) away from the coherent sources. The separation between the two coherent sources is (a) \(1.0 \mathrm{~mm}\) (b) \(0.1 \mathrm{~mm}\) (c) \(0.05 \mathrm{~mm}\) (d) \(0.01 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The separation between the coherent sources is 0.1 mm.

Step 1: Identify the Given Values

The wavelength of light, \( \lambda \), is given as \( 5000 \AA = 5000 \times 10^{-10} \mathrm{~m} \). The fringe width, \( w \), is \( 5 \mathrm{~mm} = 5 \times 10^{-3} \mathrm{~m} \). The distance to the screen, \( D \), is \( 1.0 \mathrm{~m} \).

Step 2: Apply the Fringe Width Formula

The formula for fringe width \( w \) in a double-slit interference pattern is given by: \[w = \frac{\lambda D}{d}\]where \( d \) is the separation between the coherent sources. We will rearrange this equation to solve for \( d \).

Step 3: Solve for Source Separation (d)

Rearrange the fringe width formula to find \( d \):\[d = \frac{\lambda D}{w}\]Substitute the given values:\[d = \frac{5000 \times 10^{-10} \times 1.0}{5 \times 10^{-3}}\]Calculate the value of \( d \).

Step 4: Calculate and Compare

Calculating the expression:\[d = \frac{5000 \times 10^{-10}}{5 \times 10^{-3}} = \frac{5000}{5} \times 10^{-7} = 1000 \times 10^{-7} = 0.1 \times 10^{-3} = 0.1 \mathrm{~mm}\]The correct answer from the given options is (b) \(0.1 \mathrm{~mm}\).

Key Concepts

Fringe Width

Fringe width refers to the distance between adjacent bright (or dark) bands in an interference pattern. These bands are produced during experiments involving wave phenomena, such as light. In this context, the fringe width is a critical factor in determining the characteristics of an interference pattern on a screen.
Fringe width is influenced by several factors:

• The wavelength of the light being used in the experiment.
• The distance from the light source to the screen where the pattern is displayed.
• The separation between the coherent sources, such as slits or points.

The formula to calculate fringe width (\( w \)) in a double-slit or a similar interference experiment is:\[ w = \frac{\lambda D}{d} \]where:

• \( w \) is the fringe width,
• \( \lambda \) is the wavelength of the light,
• \( D \) is the distance from the coherent sources to the screen,
• \( d \) is the separation between the coherent sources.

Fringe width provides insight into the precision and resolution of the interference pattern, key for accurately interpreting results.

Double-Slit Interference

Double-slit interference is a fundamental concept in wave physics and is essential in understanding experiments such as the biprism experiment. Discovered by Thomas Young, it demonstrates the phenomenon where waves superimpose to form alternating patterns of constructive and destructive interference. This results in bright and dark fringes on a screen.
In Young’s experiment, light passes through two closely spaced slits. These slits act as coherent sources, meaning they emit light waves that are in phase or have a constant phase difference. When these waves overlap on a screen, they create an interference pattern.
The key to understanding double-slit interference is how the waves from each slit interact:

• If the waves are in phase, they will constructively interfere to create a bright fringe.
• If the waves are out of phase, they will destructively interfere to form a dark fringe.

The double-slit experiment is vital for demonstrating the wave nature of light and provides a method for understanding wave dynamics in educational and practical applications.

Coherent Sources

Coherent sources are crucial in producing clear and distinct interference patterns. These are sources that emit light waves with a constant phase relationship, allowing the waves to maintain a stable interference pattern over time.
The significance of coherent sources in interference experiments, such as the biprism experiment, lies in:

• Ensuring that the light waves superimpose in a predictable manner, creating distinct and measurable interference fringes.
• Providing consistent wave properties, essential for reliable experimental results.

In practical terms, light from a single source is often split, using devices like biprisms or beam splitters, to produce coherent sources. This ensures that the waves are derived from the same original wavefront and thus remain coherent.

Wavelength of Light

The wavelength of light is a fundamental property that characterizes the nature of light waves. It is defined as the distance between two consecutive peaks (or troughs) of the wave. In interference experiments, the wavelength is pivotal in determining the behavior and properties of the interference pattern.
A shorter wavelength results in narrower fringe widths, while a longer wavelength leads to wider fringes. The wavelength of visible light typically ranges from about 4000 \( \AA \) (angstroms) for violet light to 7000 \( \AA \) for red light.
Understanding wavelength is important for applying the formula for fringe width accurately. It helps in correlating the observed interference pattern conditions with the theoretical predictions. By using the known wavelength of light in calculations, scientists and students can predict or verify the separation between the slits or coherent sources based on the observed fringe width and other experimental parameters.