Problem 28 - Master Resource Book in JEE Main Physics

Question

Two slits separated by a distance of \(1 \mathrm{~mm}\) are illuminated with red light of wavelength \(6.5 \times 10^{-7} \mathrm{~m}\). The interference fringes are observed on a screen place \(1 \mathrm{~m}\) from the slits. The distance between the third dark fringe and the fifth bright fringe is equal to (a) \(0.65 \mathrm{~mm}\) (b) \(1.63 \mathrm{~mm}\) (c) \(3.25 \mathrm{~mm}\) (d) \(4.88 \mathrm{~mm}\)

Step-by-Step Solution

Answer

(b) 1.63 mm

Step 1: Understanding the problem

In this problem, we need to identify the distance between the third dark fringe and the fifth bright fringe on a screen, created by light passing through two slits. The given data are the slit separation, wavelength of light, and distance from slits to the screen.

Step 2: Find the Position of the Third Dark Fringe

The formula for the position of dark fringes in a double-slit setup is given by:\[ y_{ ext{dark}} = \left(m + \frac{1}{2}\right) \frac{\lambda L}{d} \]Where:- \( d = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \), the slit separation- \( L = 1 \text{ m} \), the distance to the screen- \( \lambda = 6.5 \times 10^{-7} \text{ m} \), the wavelength of light- \( m = n - 1 \) for dark fringes, where \( n \) is the order of the dark fringeFor the third dark fringe \( (n = 3) \), we have \( m = 2 \):\[ y_{ ext{dark, 3}} = \left(2 + \frac{1}{2}\right) \frac{6.5 \times 10^{-7} \times 1}{1 \times 10^{-3}} = 1.625 \times 10^{-3} \text{ m} \] or \( 1.625 \text{ mm} \).

Step 3: Find the Position of the Fifth Bright Fringe

The formula for the position of bright fringes is:\[ y_{ ext{bright}} = m \frac{\lambda L}{d} \]Where:- \( m = n \) when \( n \) is the order of the bright fringeFor the fifth bright fringe \( (n = 5) \), we have:\[ y_{ ext{bright, 5}} = 5 \times \frac{6.5 \times 10^{-7} \times 1}{1 \times 10^{-3}} = 3.25 \times 10^{-3} \text{ m} \] or \( 3.25 \text{ mm} \).

Step 4: Calculate the Distance Between the Fringes

Now, we calculate the distance between the third dark fringe and the fifth bright fringe:\[ \Delta y = y_{ ext{bright, 5}} - y_{ ext{dark, 3}} = 3.25 \text{ mm} - 1.625 \text{ mm} = 1.625 \text{ mm} \].

Step 5: Conclude with the Correct Option

By comparing the calculated distance of \( 1.625 \text{ mm} \) to the given options, we see that option (b) \( 1.63 \text{ mm} \) is the closest match for the distance.

Key Concepts

Interference Patterns

In the fascinating world of physics, understanding interference patterns reveals the wave nature of light. When coherent light, like a laser beam, hits a pair of tiny slits, it splits into two waves that travel in slightly different paths. These waves spread out and overlap on a screen leading to an interference pattern. This pattern consists of alternating bright and dark bands, known as fringes. Bright fringes appear when the peaks of the two waves align, creating constructive interference. Conversely, dark fringes appear when the peaks of one wave align with the troughs of the other, leading to destructive interference. This interplay creates a sinusoidal pattern that shows the wave behavior of light. Viewing these patterns allows us to correlate this observable effect with the fundamental principles of wave physics, further emphasizing the dual nature of light.

Fringe Distance Calculation

Calculating fringe distances in a double-slit experiment allows you to find out how far the bands of light and dark are from each other. This involves understanding the geometry of the setup and the properties of light used.

• The formula for finding bright fringes is: \( y_{\text{bright}} = m \frac{\lambda L}{d} \)
• The formula for dark fringes is slightly different: \( y_{\text{dark}} = \left(m + \frac{1}{2}\right) \frac{\lambda L}{d} \)

In both formulas, \( y \) is the distance on the screen, \( m \) is the order number of the fringe, \( \lambda \) is the wavelength of light used, \( L \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.
For instance, finding the distance between specific fringes involves substituting the order numbers into these formulas, then subtracting the results. It's a beautiful way to directly relate theoretical physics equations to real-world observations, showing you how even abstract numbers have tangible outcomes.

Young's Double Slit Experiment

Young's Double Slit Experiment serves as a cornerstone for understanding wave interference. Conducted successfully by Thomas Young in 1801, the experiment demonstrated the wave-like nature of light. It fundamentally showed that light can exhibit interferences, much like water or sound waves.
In the setup, light from a single source is split into two coherent paths by two closely spaced slits. As these two light waves overlap on a distant screen, they produce a series of interference fringes; a direct visual representation of wave interaction. This setup is simple yet revealing—the alternating dark and bright lines aren't just pretty patterns; they're fingerprints of wave interference.
This experiment further underpins modern technologies like lasers, influencing fields from optics to quantum physics. Even today, Young's insights reinforce our understanding of light behavior, showing not only light's dual nature but also that minute variations in path lengths can result in profound changes in light's appearance on a screen, changing the landscape of physical optics forever.