Question

An electromagnetic wave going through vacuum is described by \(E=E_{0} \sin (k x-\omega t)\) then \(B=B_{0} \sin (k x-\omega t)\) then (A) \(E_{0} B_{0}=\operatorname{cok}\) (B) \(E_{0} k=B_{0} \omega\) (C) \(\mathrm{E}_{0} \mathrm{~m}=\mathrm{B}_{0} \mathrm{k}\) (D) none of these

Short answer

The correct answer is (D) none of these because the relation we derived between the amplitudes of electric and magnetic fields (\(E_0 / B_0 = c\)) does not match any of the given options.

Step-by-step solution

Understand the values of variables in the given equations

In the given equations, we have \(E = E_0 \sin(kx - \omega t)\) and \(B = B_0 \sin(kx - \omega t)\). Here, \(E_0\) and \(B_0\) are the amplitudes of the electric and magnetic fields, respectively; k is the wave vector, \(\omega\) is the angular frequency, x is the position, and t is the time.

Compare the coefficients of sin function in both given equations

Now, compare the coefficients of the sin function in both E and B: \(E=E_0 \sin(kx-\omega t) \Rightarrow E_0 = E /\sin(kx - \omega t)\) \(B=B_0 \sin(kx-\omega t) \Rightarrow B_0 = B /\sin(kx - \omega t)\)

Use the relation between the magnitudes of electric and magnetic fields and check the given options

We know the relation between magnitudes of electric and magnetic fields: \(E = cB\). From Step 2, we can write this relation in terms of the amplitudes: \(E_0 = cB_0 \Rightarrow E_0 / B_0 = c\) Now let's check the options for the relation between \(E_0\) and \(B_0\): (A) \(E_0 B_0 = cok\) ⟹ This option is not consistent with our derived relation. (B) \(E_0 k = B_0 \omega\) ⟹ From our derived relation, if k = c, then this option is correct. However, we cannot assume this, so we cannot determine whether this option is correct or not. (C) \(E_0 m = B_0 k\) ⟹ This option is not consistent with our derived relation. (D) None of these ⟹ Since our derived relation is not matching with any provided options, this option is the correct answer. Hence, the correct answer is (D) none of these.

Key concepts

Wave Equations

Electromagnetic waves are described mathematically using wave equations. These equations represent how the electric and magnetic fields vary with position and time in a wave. In this exercise, the wave equations used are given as:

• Electric Field: \(E = E_0 \sin(kx - \omega t)\)
• Magnetic Field: \(B = B_0 \sin(kx - \omega t)\)

Here, \(E_0\) and \(B_0\) are constants that represent the maximum values, or amplitudes, of the electric and magnetic fields, respectively. The terms \(kx - \omega t\) describe how these fields oscillate in both space (\(x\)) and time (\(t\)). Understanding these equations helps us analyze the behavior of electromagnetic waves in a vacuum, where they are known to travel at the speed of light.

Electric Field Amplitude

The electric field amplitude, denoted by \(E_0\), is a crucial parameter in the study of electromagnetic waves. It represents the maximum value that the electric field attains during oscillation. The amplitude \(E_0\) helps determine the intensity and energy of the wave. A higher amplitude indicates a stronger electric field, which in turn suggests the wave carries more energy.

• The relationship between the electric field \(E\) and its amplitude is: \[E = E_0 \sin(kx - \omega t)\]
• This relationship means the electric field varies sinusoidally, reaching its peak (\(E_0\)) at certain points.

This parameter is also essential when discussing how light and other electromagnetic waves interact with matter, influencing phenomena such as reflection and refraction.

Magnetic Field Amplitude

Similar to the electric field, the magnetic field in an electromagnetic wave has its amplitude, denoted by \(B_0\). This amplitude represents the maximum strength of the magnetic part of the electromagnetic wave.The magnetic field and its amplitude are intimately related to the electric field, as both are perpendicular to each other in an electromagnetic wave. The equation describing the magnetic field is:

• \(B = B_0 \sin(kx - \omega t)\)

Just like with electric fields, a larger magnetic field amplitude \(B_0\) indicates a more powerful magnetic influence by the wave. This explains how magnetic fields can induce currents and exert forces on moving charges, a fundamental principle employed in technologies such as MRI machines and wireless communication systems.

Wave Vector

The wave vector, symbolized by \(k\), is a vector quantity that provides valuable information on the propagation characteristics of a wave. This includes its direction in space and its spatial frequency—how many wave cycles exist per unit distance.The magnitude of the wave vector is directly related to the wavelength \(\lambda\) of the wave through the equation:

• \(k = \frac{2\pi}{\lambda}\)

Since \(k\) appears in the wave equations \(\sin(kx - \omega t)\), it defines how quickly the wave oscillates as it moves through space. A large wave vector suggests a short wavelength, indicating rapid oscillations.This concept is critical in measuring how electromagnetic waves travel through different media, such as glass or air, and for determining the resolving power of optical instruments like telescopes and microscopes.

Angular Frequency

Angular frequency, denoted by \(\omega\), characterizes how many wave cycles occur per unit time. It is a measure of the rate at which the wave oscillates as time progresses.The relationship between angular frequency \(\omega\) and the period \(T\) of the wave (the time for one complete cycle) is given by:

• \(\omega = \frac{2\pi}{T}\)

This parameter is pivotal in describing temporal characteristics of the wave, much like the wave vector describes spatial characteristics. For example, in the wave equations \(\sin(kx - \omega t)\), angular frequency \(\omega\) dictates how rapidly the wave oscillates with time.Understanding angular frequency is essential when studying phenomena like sound waves, light waves, and other periodic motions, as it helps in quantifying how different waves interact with time-varying systems and materials.