Problem 2118 - JEE Physics

Question

An electromagnetic wave going through vacuum is described by \(E=E_{0} \sin (k x-\cot )\). Which of the following is independent of the wavelength? (A) \(\omega\) (B) \((\mathrm{k} / \mathrm{c})\) (C) \(\mathrm{k}_{\mathfrak{e}}\) (D) \(\mathrm{k}\)

Step-by-Step Solution

Answer

None of the given options seem to be independent of wavelength. However, if we exclude option (C) as it is poorly defined in this context, we can conclude that all other options depend on the wavelength.

Step 1: Write given equation in a more readable form

First, let's rewrite the given equation in a more readable form using latex notation: \(E = E_0 \sin(kx - \omega t)\)

Step 2: Identify the relationships between variables

Wave number (k) and wavelength (λ) are related by: \(k = \frac{2\pi}{\lambda}\) Angular frequency (ω) and wave speed (c) are related as: \(\omega = ck\)

Step 3: Analyze the given options with respect to wavelength

Now let's analyze each option to see if it depends on the wavelength: (A) \(\omega\) By using the relationship between \(\omega\), c, and k, we can see that: \(\omega = c \cdot \frac{2\pi}{\lambda}\) Here, \(\omega\) depends on \(\lambda\). (B) \(\frac{k}{c}\) By using the relationship between wave number and wavelength, we can express this option in terms of wavelength: \(\frac{k}{c} = \frac{\frac{2\pi}{\lambda}}{c}\) Here, this expression depends on \(\lambda\). (C) \(\mathrm{k}_{\mathfrak{e}}\) Option (C) seems to introduce an unknown variable \(\mathrm{k}_{\mathfrak{e}}\), which is not useful in this context. We cannot judge if this is independent of wavelength or not. (D) \(k\) From the relationship between wave number (k) and wavelength (λ): \(k = \frac{2\pi}{\lambda}\) Here, k is dependent on \(\lambda\).

Step 4: Choose the correct option

Based on our analysis, none of the given options seem to be independent of wavelength, so we cannot give a definitive answer. However, if we exclude option (C) as it is poorly defined in this context, we can conclude that all other options depend on the wavelength.

Key Concepts

Wave Equation

The wave equation is a fundamental formula in physics that describes how waves propagate through space and time. It provides a way to mathematically represent the displacement of a wave, such as an electromagnetic wave, through a medium. The standard form of the wave equation for an electromagnetic wave in a vacuum is:\[ E(x, t) = E_0 \sin(kx - \omega t) \]Here:

• \( E(x, t) \) is the electric field at position \( x \) and time \( t \).
• \( E_0 \) is the amplitude of the wave, representing the maximum strength of the electric field.
• \( k \) is the wave number, indicating how many waves fit into a unit of space.
• \( \omega \) is the angular frequency, which tells us how fast the wave oscillates in time.

The wave equation combines spatial and temporal components to give a full picture of wave dynamics. Understanding this equation is crucial for analyzing how different parameters affect wave behavior.

Wave Number

The wave number \( k \) is a key concept that indicates the number of wave cycles within a unit distance. It helps determine the spatial properties of a wave. Mathematically, the wave number is related to the wavelength \( \lambda \) through the equation:\[ k = \frac{2\pi}{\lambda} \]

• This formula tells us that shorter wavelengths correspond to higher wave numbers, while longer wavelengths have lower wave numbers.
• Since \( k \) is inversely proportional to \( \lambda \), any changes in wavelength directly affect the wave number.

Wave number is an essential part of understanding the distribution of energy within a wave, and it plays a critical role in calculating wave speed and frequency.

Angular Frequency

Angular frequency \( \omega \) is a measure of how fast a wave oscillates over time. It is crucial in determining the temporal behavior of a wave. Angular frequency is related to the wave speed \( c \) and the wave number \( k \) by the equation:\[ \omega = ck \]

• This implies that angular frequency is directly proportional to both wave speed and wave number.
• If you know the wave speed and wave number, you can easily find the angular frequency.
• Since \( k \) is dependent on \( \lambda \), \( \omega \) is also influenced by changes in wavelength.

Understanding angular frequency helps to describe the cyclical nature of waves, making it fundamental in both theoretical and applied physics.

Wavelength

Wavelength \( \lambda \) is the distance between two consecutive points in phase on a wave, such as peak to peak or trough to trough. It is a fundamental property that describes the spatial periodicity of a wave. In the context of electromagnetic waves, wavelength can be calculated if you know the wave number using:\[ \lambda = \frac{2\pi}{k} \]

• Wavelength plays a significant role in determining the wave's behavior in different media.
• It affects how waves interact with obstacles and other waves, which is critical in optics and acoustics.

A thorough understanding of wavelength is essential for exploring phenomena like diffraction, interference, and the Doppler effect in wave mechanics.

Wave Speed

Wave speed \( c \) is the rate at which the wave travels through a medium. In a vacuum, electromagnetic waves travel at the speed of light, roughly \( 3.00 \times 10^8 \) meters per second. Wave speed is related to both wavelength and angular frequency through the formula:\[ c = \frac{\omega}{k} \]

• This relation implies that if you know either the angular frequency and wavelength, or the wave number, you can calculate the speed of the wave.
• For electromagnetic waves in a vacuum, the speed is constant regardless of the changes in wavelength.

Understanding wave speed is vital for designing and analyzing systems involving wave propagation, such as telecommunications, radar, and medical imaging.