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
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:
$\left( \nu _{ph}^{2}{{\nabla }^{2}}-\frac{{{\partial }^{2}}}{\partial {{t}^{2}}} \right)E=0$
$\left( \nu _{ph}^{2}{{\nabla }^{2}}-\frac{{{\partial }^{2}}}{\partial {{t}^{2}}} \right)B=0$
Where,
${{\nu }_{ph}}=\frac{1}{\sqrt{\mu \in }}$
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)
𝜙B = Magnetic flux
J = Current density
I = Electric current
c = Speed of light
μ0 = 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 dA suggests the surface area of the closed surface you’re calculating for (sometimes written as dS). 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.
Summary
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.
FAQs on Electromagnetic Wave Theory of Maxwell
1. What is the core concept of Maxwell's electromagnetic wave theory?
Maxwell's electromagnetic wave theory proposes that light is an electromagnetic wave. It states that a time-varying electric field produces a time-varying magnetic field, and vice-versa. These two oscillating fields sustain each other, propagating through space as a transverse wave at the speed of light, and do not require any material medium for travel.
2. What are the fundamental properties of an electromagnetic wave according to Maxwell?
According to Maxwell's theory, electromagnetic (EM) waves have several key properties:
- They are transverse waves, meaning the electric and magnetic fields oscillate perpendicular to each other and also perpendicular to the direction of wave propagation.
- They travel through a vacuum at a constant speed, 'c' (the speed of light), which is approximately 3 x 108 m/s.
- They do not require a physical medium to travel and can propagate through a vacuum.
- The energy of an EM wave is carried by its oscillating electric and magnetic fields.
3. How did Maxwell's theory unify electricity, magnetism, and optics?
Before Maxwell, electricity, magnetism, and optics were considered separate branches of physics. Maxwell's genius was in synthesising the work of Gauss, Faraday, and Ampere into a single, cohesive set of four equations. By adding his concept of displacement current, his equations predicted the existence of waves of oscillating electric and magnetic fields. When he calculated the speed of these waves, it turned out to be the same as the measured speed of light. This was the first strong evidence that light itself is an electromagnetic wave, thus unifying three previously distinct fields into the single theory of electromagnetism.
4. What are Maxwell's four equations and what does each one represent?
Maxwell's four equations are the foundation of classical electromagnetism. In simple terms, they are:
- Gauss's Law for Electricity: Describes how electric fields originate from electric charges. It states that electric field lines begin on positive charges and end on negative charges.
- Gauss's Law for Magnetism: States that there are no magnetic monopoles. Magnetic field lines are continuous loops without a beginning or an end.
- Faraday's Law of Induction: Explains how a changing magnetic field induces an electromotive force (EMF), which in turn creates an electric field.
- Ampere-Maxwell Law: Describes how magnetic fields are created by either an electric current or a changing electric field (the displacement current). This final piece was Maxwell's key contribution that predicted EM waves.
5. Why is the concept of 'displacement current' so important in Maxwell's theory?
The displacement current was Maxwell's crucial addition to Ampere's Circuital Law. Before Maxwell, Ampere's law only explained magnetic fields produced by moving charges (conduction current). However, it failed in situations with changing electric fields, such as in the gap of a charging capacitor. Maxwell proposed that a changing electric flux in the gap acts as a source of the magnetic field, just like a real current. This 'displacement current' is essential because it shows that a changing electric field can create a magnetic field, which is the key mechanism that allows electromagnetic waves to propagate through empty space.
6. What are some common examples of technologies based on Maxwell's electromagnetic wave theory?
Almost all modern communication and wireless technologies are direct applications of Maxwell's theory. Key examples include:
- Wireless Communication: Mobile phones, Wi-Fi, and Bluetooth all use radio waves (a type of EM wave) to transmit information.
- Broadcasting: Radio and television signals are transmitted as electromagnetic waves.
- Radar Systems: Used in aviation and weather forecasting, radar works by sending out microwaves and analysing the reflected waves.
- Medical Imaging: X-rays and Magnetic Resonance Imaging (MRI) use different parts of the electromagnetic spectrum.
- Microwave Ovens: Use microwaves to transfer energy to water molecules in food, heating it up.
7. If Maxwell's theory was so successful, why couldn't it explain phenomena like the photoelectric effect?
While Maxwell's theory perfectly describes the wave nature of light on a macroscopic scale, it has limitations at the atomic and subatomic levels. It could not explain certain experimental observations that emerged in the late 19th and early 20th centuries. These included:
- The photoelectric effect, where light ejects electrons from a material, which depends on frequency, not intensity (a wave property).
- Black-body radiation, which describes the spectrum of light emitted by a hot object.
- The discrete line spectra of atoms.
These phenomena could only be explained by treating light not as a continuous wave, but as discrete packets of energy called 'quanta' or 'photons', which led to the development of quantum mechanics.
