Problem 107 - Master Resource Book in JEE Main Physics

Question

\mathrm{\\{} T h i s ~ q u e s t i o n ~ h a s ~ a ~ p a r a g r a p h ~ f o l l o w e d ~ b y ~ t w o ~ statements, Statement 1 and Statement 2. Of the given four alternatives after the statements, choose the one that describes the statements. A thin air film is formed by putting the convex surface of a plano-convex lens over a plane glass plate. With monochromatie light, this film gives an interference pattern due to light reflected from the top (convex) surface and he bottom (glass plate) surface of the film. Statement 1 When light reflects from the air-glass plate interface, the reflected wave suffers a phase change of \(\pi\). Statement 2 The centre of the interference pattern is dark. (a) Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation of Statement \(\mathbb{I}\). (b) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement \(1 .\) (c) Statement 1 is false, Statement 2 is true. (d) Statement 1 is true, Statement 2 is false.

Step-by-Step Solution

Answer

Option (a): Both statements are true, and Statement 2 correctly explains Statement 1.

Step 1: Understanding the Interference Pattern

When a convex lens is placed on a plane glass plate, an air film forms, and interference is due to light reflected from two surfaces: the top of the air film (convex lens) and the bottom (glass plate). The key to this problem is understanding the phase changes that occur upon reflection.

Step 2: Analyze Statement 1

Statement 1 refers to the phase change upon reflection at an air-glass interface. When light reflects off a denser medium like glass, the wave undergoes a phase change of \(\pi\) radians (180 degrees). Thus, Statement 1 is true.

Step 3: Analyze Statement 2

Statement 2 claims that the center of the interference pattern is dark. Due to the \(\pi\) phase shift at the glass interface, destructive interference occurs at the thinnest point (center of the pattern), making the center dark. Thus, Statement 2 is true.

Step 4: Determine the Relationship Between the Statements

Both statements are true independently. Statement 2 explains why the center is dark due to the \(\pi\) phase change accounted for in Statement 1. Therefore, Statement 2 properly explains Statement 1.

Key Concepts

Phase Change in Optics

When light encounters different media, it undergoes a phase change, which is especially interesting in optics. Imagine light as a wave—when it hits a boundary between two different materials, such as air and glass, its behavior changes. This change often involves a shift in phase, the wave's oscillating pattern.
This is crucial for two types of reflections:

• At an air-glass interface, like in our exercise, light striking the glass from air experiences a phase change of \(\pi\) radians, equivalent to 180 degrees.
• Conversely, light reflecting off an object less optically dense than its current medium does not undergo this phase shift.

Understanding this change helps us predict how light waves create patterns of interference by combining waves with either constructive or destructive effects.

Interference Patterns

Interference patterns arise when two or more light waves overlap. These patterns can vary, creating beautiful bands of light or darkness based on how the waves interact. In our exercise, this phenomenon is observed using a thin air film between a lens and glass plate. Light reflects off both surfaces, leading to two important scenarios:

• Constructive Interference: When waves are perfectly in phase, their amplitudes add up, causing bright areas.
• Destructive Interference: When out of phase by \(\pi\) radians, they cancel each other out, leading to dark spots.

The central dark spot observed in interference patterns is a classic example of destructive interference, caused by a \(\pi\) phase shift along the air-glass interface.

Reflection at Air-Glass Interface

Reflecting light at an air-glass interface is where the subtleties of optics shine. The air film created by placing a convex lens on a flat glass plate causes light to reflect from two main points: the top surface of the lens and the glass below.
When light hits the glass surface, it sees a denser medium and flips its phase by \(\pi\) radians. This phase shift is crucial as it directly influences the resulting pattern of the interference. Knowing that the light only shifts phase in one of the reflections helps in predicting which parts of the interference pattern will appear dark or light.
This concept of phase change helps determine why patterns such as rings or other shapes appear in thin film interference, often highlighted in monochromatic light for clearer observation.