Deriving the Magnetic Field Intensity H from E and B in Electromagnetic Waves: A Comprehensive Guide

The process of deriving the magnetic field intensity H from the electric field E and the magnetic flux density B in electromagnetic waves involves the use of Maxwell's equations and the wave equation. This guide provides a step-by-step explanation to understand this intricate and fundamental process.

Maxwell's Equations and Their Relevance

Maxwell's equations are a set of four differential equations that describe how electric and magnetic fields interact. These equations are foundational in the study of electromagnetism. The relevant ones for deriving the wave equations are:

1. Faraday's Law of Electromagnetic Induction

u0393B - u03D8B/DISABLED u03C4

This law states that the electromotive force (EMF) induced in a closed circuit is proportional to the rate of change of magnetic flux through the circuit. This principle is crucial for understanding how changes in the magnetic field can induce an electric field.

2. Ampère-Maxwell Law

u0393B u03BCoJo u03D8E/DISABLED u03C4

This modified law introduces the displacement current, accounting for changes in the electric field that can induce a magnetic field, even in the absence of conventional currents. This is a critical concept for understanding electromagnetic waves in the absence of electric charges.

The Wave Equation in Electromagnetic Waves

The wave equation is a second-order partial differential equation that describes the propagation of a variety of waves, including sound waves, light waves, and water waves. For electromagnetic waves, the wave equations can be derived from Maxwell's equations and are given by:

Wave Equation for Electric Field E

u22072E u03BCou03D5E u22C5 E/DISABLED u03C42

Wave Equation for Magnetic Field B

u22072B u03BCou03D5E u22C5 B/DISABLED u03C42

These wave equations describe how the electric and magnetic fields propagate through space and time, forming electromagnetic waves.

Derivation of the Wave Equations

To derive the wave equations from Maxwell's equations, we take the curl of both sides of Faraday's Law and substitute the relevant terms from the Ampère-Maxwell Law. Similarly, we take the curl of both sides of the Ampère-Maxwell Law and substitute the relevant terms from Faraday's Law.

The detailed mathematical steps involve:

Taking the curl of Faraday's Law: u2207 u22A5 u0393B u03D8E/DISABLED u03C4 Substituting the Ampère-Maxwell Law into the equation: u2207 u22A5 0B u03BCoJo u03D8E/DISABLED u03C4 Simplifying to derive the wave equation for the electric field: u22072E u03BCou03D5E u22C5 E/DISABLED u03C42

The same steps are followed to derive the wave equation for the magnetic field.

Relation Between E, B, and H

In a source-free region of free space, the magnetic field intensity H is related to the magnetic flux density B by the equation:

H B/DISABLED u03BCo

where DISABLED u03BCo is the permeability of free space. Once B is obtained from the wave equation, the magnetic field intensity H can be calculated using this relation.

It is important to note that this derivation assumes that the fields are in a source-free region, meaning there are no electric charges or currents present. The presence of such charges and currents would modify Maxwell's equations and the resulting wave equations.

This comprehensive guide aims to provide a clear and detailed explanation of the process of deriving the magnetic field intensity H from E and B in electromagnetic waves, using the principles of Maxwell's equations and the wave equation.