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Electromagnetic Wave Theory of Maxwell

Last updated date: 29th May 2024
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A Brief Overview of Maxwell’s Theory

Maxwell not only developed an entire electromagnetic theory, represented by Maxwell's equations but brought along all the work that had been done by good physicists like Oersted, Coulomb, Gauss, and Michael Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations include the most important laws of electricity and magnetism.

Maxwell calculated the speed of an electromagnetic wave and found that the speed of an electromagnetic wave was almost identical to the speed of light. In this article, we will discuss the electromagnetic theory of Maxwell.

What is Maxwell’s Electromagnetic Wave Theory?

According to Maxwell’s EM wave theory , light waves are related to changing electric fields and magnetic fields. The change within the electrical and magnetic field leads to the propagation of electromagnetic waves or light waves. The main points of this theory are:

  • The energy is emitted from any source continuously in the form of radiation and is termed radiant energy.

  • The radiation includes electrical and magnetic fields oscillating perpendicular to each other and both perpendicular to the direction of propagation of the radiation.

  • Radiations possess wave characteristics and travel with the speed of light.

  • Radiations are known as electromagnetic radiations or electromagnetic waves.

  • These waves don't need any material medium for propagation.

Electromagnetic Wave

An electromagnetic wave consists of an electric field wave and a magnetic field wave oscillating back and forth, aligned at right angles to each other. The oscillation of the electrical part of the wave generates the magnetic field, and also the oscillating of this part successively produces an electrical field again, on and on because it travels through space.

Electromagnetic wave

Electromagnetic Wave

Like any other wave, electromagnetic radiation includes a frequency and a wavelength, and the product of these is always equal to ​c​, the speed of light. Electromagnetic waves are all around us, and as well as visible radiation, other wavelengths are usually known as radio waves, microwaves, infrared, ultraviolet, X-rays, and gamma rays. All of these types of electromagnetic radiation have a similar basic kind as explained by Maxwell’s equations; however, their energies vary with frequency (i.e., a higher frequency means that higher energy).

Electromagnetic Wave Equation

The equation of electromagnetic wave explains the transmission of electromagnetic waves in a vacuum or over a medium.

The EM wave equation could be a second-order fractional differential equation. It's a three-dimensional kind of differential equation. The standardised form of the equation is written as:

$\left( \nu _{ph}^{2}{{\nabla }^{2}}-\frac{{{\partial }^{2}}}{\partial {{t}^{2}}} \right)E=0$

$\left( \nu _{ph}^{2}{{\nabla }^{2}}-\frac{{{\partial }^{2}}}{\partial {{t}^{2}}} \right)B=0$


${{\nu }_{ph}}=\frac{1}{\sqrt{\mu \in }}$

Maxwell’s Equations

Electric field lines originate on positive charges and terminate on negative charges. The electrical field is defined as the force per unit charge on a test charge, and additionally, the strength of the force is associated with the electrical constant, also called the free space permittivity.

From Maxwell’s 1st equation we have a tendency to acquire a special form of Coulomb’s law referred to as Gauss’s law for electricity.

Differential form, $\nabla \cdot E=\frac{\rho }{{{\in }_{0}}}$

Integral Form, $\int{E\cdot }dA=\frac{q}{{{\in }_{0}}}$

Magnetic field lines are continuous, having no starting or finishing point. No magnetic monopoles are known to exist. The strength of the magnetic force is related to the magnetic constant, also referred to as the permeability of free space. This second of Maxwell’s equations is understood by Gauss’s law for magnetism.

Differential form, $\nabla \cdot B=0$

Integral form, $\int{B\cdot dA=0}$

A changing magnetic flux induces an electromotive force (emf) and hence an electrical field. This third of Maxwell’s equations is called Faraday’s law of induction.

Differential form, $\nabla \times E=-\frac{\partial B}{\partial t}$

Integral form, $\int{E\cdot ds=-\frac{{{\partial }_{\in B}}}{\partial t}}$

Magnetic fields are generated by moving charges or by changing electrical fields. This fourth of Maxwell’s equations encompasses Ampere’s law and adds another supply of magnetism—changing electrical fields.

Differential form, $\nabla \times B=\frac{J}{{{\in }_{0}}{{c}^{2}}}+\frac{1}{{{c}^{2}}}\frac{\partial E}{\partial t}$

Integral form, $\int{B\cdot ds={{\mu }^{0}}}I+\frac{1}{{{c}^{2}}}\frac{\partial }{\partial t}\int{E\cdot dA}$

Symbols Used in Maxwell’s Equations

Maxwell’s equations use a fairly wide range of symbols, and it’s necessary to understand what these mean if you’re aiming to learn to use them. Therefore, here’s a run-down of the meanings of the symbols used:

  • ​B​ = Magnetic field

  • ​E​ = Electric field

  • ​ρ​ = Electric charge density

  • ​ε0​ = Permittivity of free space

  • ​q​ = Total electric charge (net total of positive charges and negative charges)

  • ​𝜙​B = Magnetic flux

  • ​J​ = Current density

  • ​I​ = Electric current

  • ​c​ = Speed of light

  • ​μ​0 = Permeability of free space = 4π × 10−7 N / A2

Additionally, it’s necessary to understand that 𝛻 is the del operator, a dot between 2 quantities (​​X​ ∙ ​Y​​) shows a scalar product, and a bolded multiplication symbol between 2 quantities could be a vector product (​​X​ × ​Y​​). Finally, the ​​A​​ in d​​A​​ suggests the surface area of the closed surface you’re calculating for (sometimes written as d​​S​​). Also, the s in the ds is a very tiny part of the boundary of the open surface you’re calculating for.

Limitations of Electromagnetic Wave Theory

This theory couldn't explain the following:

  • The phenomena of black body radiation

  • The photoelectric effect

  • The variation of heat capacity of solid as a function of temperature

  • The line spectra of atoms with special reference to H

Interesting Facts

  • Maxwell created fundamental contributions to the development of physical science.

  • He was also the founder of the kinetic theory of gases.

  • This theory provided the new subject of applied mathematics, and physics, linking thermodynamics and mechanics, and is still widely used as a model for rarefied gases and plasmas.

Important Questions

Q1. Where are Maxwell equations used?

Ans. The equations give a mathematical model for electrical, optical, and radio technologies, like power generation, electrical motors, wireless communication, lenses, radar, etc. They describe how electrical and magnetic fields are generated by charges, currents, and changes in the fields.

Q2. What is the importance of Maxwell theory?

Ans. This Maxwell theory provided the new subject of statistical physics, linking thermodynamics and mechanics, and is still widely used as a model for rarefied gases and plasmas.


Maxwell’s equations illustrate how apparently straightforward mathematical statements can elegantly unite and express a large number of concepts—why arithmetic is the language of science. Maxwell’s equations include the most important laws of electricity and magnetism. The derivation and the symbols used in the Maxwell equation can be understood through this article and the limitations of electromagnetic wave theory can be explained.

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FAQs on Electromagnetic Wave Theory of Maxwell

1. State Faraday’s law of electromagnetic induction.

Maxwell's third equation comes from Faraday's laws of electromagnetic induction. Faraday’s law is the basic law of electromagnetism predicting how magnetic flux can interact with an electrical circuit to provide electromagnetism (EMF) – a phenomenon called electromagnetic induction. Faraday’s first law of electromagnetic induction states that whenever the flux of magnetic field through the area bounded by a closed-loop system changes, an electromotive force is created in the loop. Faraday’s second law of electromagnetic induction states that the magnitude of induced electromotive force is directly proportional to the time rate of change in magnetic flux connected with the circuit.

2. What is electromagnetic induction?

Electromagnetic induction is the phenomenon where whenever the magnetic flux coupled with an electrical circuit changes, an e.m.f. is induced within the circuit. The magnitude of induced emf is directly proportional to the speed of change in magnetic flux. The S.I. the unit of magnetic flux is weber/tesla-metre. The magnetic flux through any surface placed in a magnetic field is the total value of magnetic lines of force crossing the field the total number of magnetic lines of force crossing the surface is generally, denoted by ϕ or ϕB.

3. State the applications of Maxwell’s equation. 

The Maxwell equations offer a mathematical model for electrical, optical, and radio technologies, like power generation, electrical motors, wireless communication, lenses, radar, etc. They describe how electrical and magnetic fields are generated by charges, currents, and changes within the fields. Maxwell's equations offer us the concept that a changing magnetic flux continuously induces an electrical field and a changing field always induces a magnetic field. Maxwell's equations are four partial differential equations that form the foundation of electromagnetic theory.