Problem 53 - Master Resource Book in JEE Main Physics

Question

In an interference pattern produced by two identical slits, the intensity at the slit of the central maximum is \(I\). The intensity at the same spot when either of the slits is closed is \(I_{0}\). Therefore, (a) \(I=I_{0}\) (b) \(l=2 I_{0}\) (c) \(l=4 l_{0}\) (d) \(I\) and \(I_{0}\) are not related to each other

Step-by-Step Solution

Answer

Option (c) \(I = 4 I_0\) is the correct answer.

Step 1: Understanding Intensity with Both Slits Open

When both slits are open, they produce an interference pattern, with maximum intensity at the central peak due to constructive interference. The intensity at the central maximum is given as \(I\). This happens when the waves from both slits constructively interfere perfectly.

Step 2: Understanding Intensity with One Slit Closed

When one of the slits is closed, there is no interference pattern. The intensity observed is simply the intensity of the wave passing through the single open slit, which is denoted as \(I_0\).

Step 3: Relating Intensity for Both Slits Open to One Slit Closed

In interference patterns, the intensity \(I\) at the central maximum when both slits are open is actually twice the sum of the intensity from each slit, assuming perfect interference. Thus, \(I = 4I_0\) because each individual intensity \(I_0\) contributes to the combined intensity with a factor of 2 due to constructive interference.

Step 4: Choosing the Correct Option

Given the relationship derived, \(I = 4I_0\). Therefore, option (c) \(I = 4 I_0\) is the correct choice.

Key Concepts

Interference Pattern

In the fascinating world of optics, an interference pattern arises when waves overlap and interact with each other. This usually happens when light waves pass through two or more slits, creating a series of dark and bright fringes on a screen. These patterns are the visible result of constructive and destructive interference at different points.

• Bright bands appear where the waves reinforce each other, known as constructive interference.
• Dark bands emerge when the waves cancel each other out, referred to as destructive interference.

This pattern can be observed in the classic double slit experiment, a pivotal demonstration in understanding wave behavior. The positions of these fringes depend on factors like the wavelength of the light and the separation of the slits. By analyzing these patterns, scientists discovered key insights into the wave nature of light. It’s this very pattern that helps us explore fundamental concepts in quantum physics and other scientific fields.

Constructive Interference

Constructive interference is a key phenomenon in wave physics, where two or more waves meet and their amplitudes add together, leading to increased wave intensity. This occurs under specific conditions, particularly when the crest of one wave aligns with the crest of another, resulting in a new wave pattern with a larger amplitude.
In the context of the double slit experiment, constructive interference leads to bright fringes on the interference pattern as the waves emerging from the slits combine at certain points to produce enhanced brightness. The condition for constructive interference is that the path difference between the two waves reaching a point is an integer multiple of the wavelength, represented mathematically by \(\Delta x = m \lambda\), where \(m\) is an integer.
Constructive interference is one of the core components that explain why we see alternating bright and dark bands in interference experiments, reflecting the periodic nature of light waves as they constructively bolster one another.

Intensity in Double Slit Experiment

The double slit experiment is renowned for demonstrating the wave nature of light, especially through its intensity patterns. When both slits are open, the intensity of light at any given point on the screen varies due to interference. The central maximum, typically the brightest part of the pattern, is where constructive interference is at its peak.

• When both slits are open, and perfect constructive interference occurs, the intensity at the central maximum is maximized.
• The intensity there is represented by \(I\).
• If only one slit is open, the intensity becomes \(I_0\).

For the central maximum with two slits open, the intensity relationship can be expressed as \(I = 4I_0\). This result highlights how the intensity quadruples due to the combined effect of both slits contributing equally to the overall intensity—effectively doubling the wave contributions from each slit and then further enhanced by constructive interference. This quadrupling is crucial for comprehending how interference can drastically affect light intensity in such experiments.