Problem 66 - Master Resource Book in JEE Main Physics

Question

The widths of two slits in YDSE are in the ratio \(1: 4\). The ratio of amplitudes of light waves from two slits will be (a) \(9: 1\) (b) \(4: 1\) (c) \(1: 2\) (d) \(1: \sqrt{2}\)

Step-by-Step Solution

Answer

The ratio of amplitudes is \(1:2\).

Step 1: Understanding the Relation Between Slit Width and Amplitude

In Young's Double Slit Experiment (YDSE), the intensity of light coming from a slit is proportional to the area of the slit, and hence the width of the slit affects the amplitude. Specifically, the amplitude of the light wave, denoted as \( A \), from a slit is proportional to the square root of the width of the slit. Thus, if \( w_1 \) and \( w_2 \) are widths of the slits, the amplitudes \( A_1 \) and \( A_2 \) are proportional to \( \sqrt{w_1} \) and \( \sqrt{w_2} \), respectively.

Step 2: Determine the Ratio of Amplitudes

The given slit width ratio is \( w_1 : w_2 = 1 : 4 \). Using the relationship that the amplitude is proportional to the square root of the width, we find the amplitude ratio \( A_1 : A_2 = \sqrt{w_1} : \sqrt{w_2} \). Substitute for the widths: \( A_1 : A_2 = \sqrt{1} : \sqrt{4} = 1 : 2 \).

Key Concepts

Amplitude of Light Waves

In Young's Double Slit Experiment, understanding the amplitude of light waves is crucial to grasping how light behaves. The amplitude of a light wave corresponds to the maximum extent of a wave from its rest position, and it determines the wave's energy. In essence, higher amplitude means higher light intensity. The relationship between slit width and amplitude is fundamental in the experiment.

• The amplitude of a light wave passing through a slit is directly related to the slit’s width.
• The formula for the amplitude (\(A\)) is proportional to the square root of the slit’s width (\(w\)): \(A \propto \sqrt{w}\).
• This means if you know the width ratio of two slits, you can determine their amplitude ratio through square root calculation.

Considering how amplitude influences light intensity, any change in slit width directly affects the wave's amplitude, which is a key concept in interference studies.

Slit Width Ratio

The slit width ratio in Young’s Double Slit Experiment is an important factor when examining light interference patterns. The ratio influences how much light intensity each slit contributes to the final interference pattern visible on a screen. The widths given here are in the ratio of \(1:4\).

• The slit width ratio defines the proportion between the widths of two slits, allowing for predictions about their relative impacts.
• In our given exercise, with widths \(w_1\) and \(w_2\) at a ratio of \(1:4\), the effect of each slit becomes evident.
• A larger slit allows more light through, increasing the light’s amplitude and intensity from that slit more than the smaller slit.

Upon seeing a slit width ratio, understanding its square root influence on amplitude can lead to predictions about light distribution in interference patterns, highlighting the experiment’s beauty and simplicity.

Light Intensity in Interference

Light intensity in Young's Double Slit Experiment results from the superposition of light waves emanating from the slits. The light intensity at a point on the interference pattern is affected by several factors, including the amplitude of light waves and phase difference. In terms of this exercise, the interplay of different factors determines the intensity of the resultant pattern.

• The intensity of light is proportional to the square of the amplitude, \(I \propto A^2\).
• Constructive interference occurs when wave amplitudes combine to increase intensity, whereas destructive interference reduces it.
• The intensity pattern shown on a screen is a result of these alternating constructive and destructive interferences.

The ability to predict and calculate light intensity at various points in an interference pattern is a testament to the understanding of wave behavior. With amplitude and slit width directly impacting this, students get a glimpse into the intricacies of light wave phenomena in this classic physics experiment.