Nature and Sources of Electromagnetic Waves

Electromagnetic Waves

Electromagnetic waves are one of the most fundamental phenomena in physics, encompassing various forms of radiation like light, radio waves, and X-rays. They are a consequence of the interplay between changing electric and magnetic fields, as described by Maxwell's equations. Unlike mechanical waves (like sound waves or water waves), which require a medium to propagate, electromagnetic waves can travel through vacuum.

---

Sources Of Electromagnetic Waves (Accelerated Charges)

The origin of electromagnetic waves lies in the behaviour of electric charges. While static charges produce electric fields and steady currents produce both static electric and magnetic fields, it is the accelerating electric charges that are the sources of electromagnetic waves.

Whenever an electric charge accelerates (changes its velocity, either in speed or direction), it radiates energy in the form of electromagnetic waves. Some examples of accelerating charges include:

• Oscillating Charges: An electric charge oscillating harmonically (like in an LC oscillator circuit or an antenna connected to an AC source) undergoes acceleration. This creates changing electric and magnetic fields that propagate outwards as electromagnetic waves. This is how radio waves are generated.
• Electrons in Atoms: When electrons transition between energy levels in atoms, they accelerate and emit photons, which are quanta of electromagnetic radiation (light, UV, X-rays).
• Electrons Striking a Target: High-speed electrons decelerating rapidly when striking a metal target produce X-rays. This is called bremsstrahlung radiation.
• Synchrotron Radiation: Charged particles accelerated in circular paths in synchrotrons emit electromagnetic radiation.

The energy carried by the electromagnetic wave comes from the kinetic energy of the accelerating charge or from the energy supplied by whatever causes the acceleration.

---

Nature Of Electromagnetic Waves ($ c = 1/\sqrt{\mu_0\epsilon_0} $, $ E_0 = cB_0 $)

Maxwell's theory predicted the existence of electromagnetic waves and described their nature. Electromagnetic waves are characterized by the following properties:

1. Composed of Oscillating Fields: An electromagnetic wave consists of mutually perpendicular, oscillating electric fields ($\vec{E}$) and magnetic fields ($\vec{B}$). These fields are also perpendicular to the direction of propagation of the wave.

   Structure of a plane electromagnetic wave propagating in the x-direction. $\vec{E}$ is in the y-direction, $\vec{B}$ is in the z-direction.

2. Self-Sustaining Propagation: A changing electric field produces a changing magnetic field (due to the displacement current term in Maxwell-Ampere Law), and a changing magnetic field produces a changing electric field (due to Faraday's Law). This continuous interplay between the changing fields allows the wave to propagate through space without needing a medium. One field variation acts as the source for the other.
3. Transverse Nature: The oscillations of the electric and magnetic fields are perpendicular to the direction in which the wave travels. This makes electromagnetic waves transverse waves. The direction of propagation is given by the direction of the vector $\vec{E} \times \vec{B}$.
4. Speed of Propagation: Maxwell's equations predicted that electromagnetic waves propagate in vacuum at a speed given by the formula:

   $ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} $

   Where $\mu_0$ is the permeability of free space and $\epsilon_0$ is the permittivity of free space.

   Substituting the values of $\mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A$ and $\epsilon_0 = 8.854 \times 10^{-12} \, C^2/(N \cdot m^2)$:

   $ c = \frac{1}{\sqrt{(4\pi \times 10^{-7} \, N/A^2) \times (8.854 \times 10^{-12} \, C^2/(N \cdot m^2))}} $

   $ c \approx \frac{1}{\sqrt{(1.2566 \times 10^{-6}) \times (8.854 \times 10^{-12})} \, (s^2/m^2)^{1/2}} $

   $ c \approx \frac{1}{\sqrt{11.12 \times 10^{-18}}} \, m/s \approx \frac{1}{3.335 \times 10^{-9}} \, m/s \approx 3.00 \times 10^8 \, m/s $

   This calculated speed is approximately equal to the speed of light in vacuum. This led Maxwell to propose that visible light is a form of electromagnetic wave. The speed of light in vacuum is a universal constant, approximately $3 \times 10^8 \, m/s$. In a medium with permeability $\mu$ and permittivity $\epsilon$, the speed of the electromagnetic wave is $v = 1/\sqrt{\mu\epsilon} = 1/\sqrt{\mu_r \mu_0 \epsilon_r \epsilon_0} = c/\sqrt{\mu_r \epsilon_r}$. The refractive index of the medium is $n = c/v = \sqrt{\mu_r \epsilon_r}$. For non-magnetic materials ($\mu_r \approx 1$), $n \approx \sqrt{\epsilon_r}$.
5. Relationship between E and B field Magnitudes: In a vacuum, the instantaneous magnitudes of the electric and magnetic fields in an electromagnetic wave are related by:

   $ E = cB $

   This relationship holds for the peak values ($E_0 = cB_0$) and the RMS values ($E_{rms} = cB_{rms}$). The electric field is typically much stronger than the magnetic field in terms of units (V/m vs T), but their effect on charged particles depends on their velocity.
6. Energy and Momentum Carrying: Electromagnetic waves carry energy and momentum. The energy flow per unit area per unit time is described by the Poynting vector ($\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$). The momentum carried by the wave results in radiation pressure when the wave is absorbed or reflected by a surface.
7. Linear Superposition: Electromagnetic waves obey the principle of linear superposition, meaning that multiple waves can exist in the same region of space and their fields simply add up vectorially.

---

Answer:

Given:

Electric field, $\vec{E}(x,t) = E_0 \sin(kx - \omega t) \, \hat{j}$

Direction of propagation is along the +x direction ($\hat{i}$).

The electric field is oscillating in the +y direction ($\hat{j}$).

In an electromagnetic wave, the electric field, magnetic field, and direction of propagation are mutually perpendicular. The direction of propagation is given by $\vec{E} \times \vec{B}$.

Here, direction of propagation is $\hat{i}$. Direction of $\vec{E}$ is $\hat{j}$. So, $\hat{j} \times (\text{direction of } \vec{B}) = \hat{i}$. This implies that the direction of $\vec{B}$ must be $\hat{k}$ (since $\hat{j} \times \hat{k} = \hat{i}$). The magnetic field is oscillating in the +z direction.

The instantaneous magnitudes are related by $E = cB$. The wave is in vacuum, so $c = \omega/k = (3 \times 10^8 \, rad/s) / (1.0 \, rad/m) = 3 \times 10^8 \, m/s$. This confirms the speed of light in vacuum.

The magnitude of the magnetic field $B_0 = E_0 / c$.

$ B_0 = \frac{50 \, V/m}{3 \times 10^8 \, m/s} = \frac{50}{3} \times 10^{-8} \, T \approx 16.67 \times 10^{-8} \, T = 1.667 \times 10^{-7} \, T $

The magnetic field oscillates in phase with the electric field, but in the z-direction. So the argument of the sine function is the same.

The expression for the magnetic field component is:

$ \vec{B}(x,t) = B_0 \sin(kx - \omega t) \, \hat{k} $

Substitute the value of $B_0$:

$ \vec{B}(x,t) = (1.667 \times 10^{-7} \, T) \sin(1.0x - 3 \times 10^8 t) \, \hat{k} $

The magnetic field is approximately $1.667 \times 10^{-7} \, T$ in amplitude, oscillating in the +z direction, in phase with the electric field.