Maxwell's Equations - Introduction 

By on

Image
James Clerk Maxwell
James Clerk Maxwell, born in
Edinburgh Scotland 1831 is renowned
for his work on the set of equations
named after him.  Maxwell published
an early form of his equations between
1861 and 1862 (a total of twenty
which were later combined into four
by Oliver Heaviside and Heinrich Hertz).

Maxwell's Equations are a set of fundamental relationships, which govern how electric and magnetic fields interact. The equations explain how these fields are generated and interact with each other, as well as their relationship to charge and current.  They form the backbone of much of modern electrical and telecommunication technology and are often quoted as being the most important equations of all time. 

The equations consist of a set of four - Gauss's Electric Field Law, Gauss's Magnetic Field Law, Faraday's Law and the Ampere Maxwell Law.  This note explains the idea behind each of the four equations, what they are trying to accomplish and give the reader a broad overview to the full set of equations.

Follow on notes will deal with each of these equations separately.  These follow on notes, will explain the mathematics (both differential and integral forms) of each law and give example applications.

Follow on notes will deal with each of these equations separately.  These follow on notes, will explain the mathematics (both differential and integral forms) of each law and give example applications.

Gauss's Electric Field Law

Gauss's Electrical law defines the relation between charge ("Positive" & "Negative") and electric field.  The law was initially formulated by Carl Friedrich Gauss in 1835.

In Gauss's law, the electric field is the electrostatic field.  It shows how the electrostatic field behaves and varies depending on the charge distribution within it.  More formally it relates the electric flux [of the electric field flowing from positive to negative charges] passing through a closed surface to the charge contained within the surface.

The electric field flux passing through a closed surface is proportional to the charged contained within that surface.

\oint_S {\vec E \cdot \hat n\,da} = \frac{{{q_{enc}}}}{{{\varepsilon _0}}}  - integral Form

\vec \nabla \cdot \vec {\rm E} = \frac{\rho }{{{\varepsilon _0}}}  - differential Form

To understand how the equations work and see application examples, please see the following note:

Gauss's Magnetic Field Law

The magnetic law defines the relationship between magnetic poles and the magnetic field. 

It is similar to the electric field law, except that individual magnetic poles ("North" & "South") cannot exist alone (whereas individual charge units can be separated from each other).  That magnetic poles must exist as a pair has a large effect on the behavior of the magnetic flux.

The total magnetic flux passing through a surface is equal to zero.

\oint_S {\vec B \cdot \hat n\,da = 0} - integral form

\vec \nabla \cdot \vec B = 0 - differential form

To understand how the equations work and see application examples, please see the following note:

  • Maxwell's Equations - Gauss's Magnetic Field Law - yet to be written

Faraday's Law

It was Michael Faraday, who in 1831 demonstrated through a series of experiments that a conductor enclosing a varying magnetic field will have a current induced.  This in turn, will create an electric field. Transformers and electrical machines all operate on the basis of Faraday's Law.

Another way to view this is that a time varying magnetic field is always associated with a time varying electric field.  And in reverse, with a time varying electric field always associated with a varying magnetic field.

A changing magnetic flux  within a surface induces a electromotive force (EMF) and induced current.  The induced current creates a circulating electric field.

\oint_C {\vec E \cdot \,d\vec l = - \frac{d}{{dt}}\oint_S {\vec B \cdot \hat n} \,da} - integral form

\vec \nabla \times \vec E = - \frac{{\partial \vec B}}{{\partial t}} - differential form

To understand how the equations work and see application examples, please see the following note:

  • Maxwell's Equations - Faraday's Law - yet to be written

Ampere Maxwell Law

Discovered in 1826 by  André-Marie Ampère, the law describes the magnetic fields produced by a varying electric current. Ampère initially considered steady electric currents and this was later expanded by Maxwell to included time varying currents.

Changing current or electric flux through a surface produces a changing magnetic field.

\oint_C {\vec B \cdot d\vec l = {\mu _0}\left[ {{I_{enc}} + {\varepsilon _0}\frac{d}{{dt}}\oint_S {\vec E \cdot \hat n\,da} } \right]} - integral form

\vec \nabla \times \vec B = {\mu _0}\left[ {\vec J + {\varepsilon _0}\frac{{\partial \vec E}}{{\partial t}}} \right] - differential form

To understand how the equations work and see application examples, please see the following note:

  • Maxwell's Equations - Ampere Maxwell Law - yet to be written

Maths Review

It is assumed that the reader is familiar with the necessary maths to be able to apply the equations.  If you need to learn or brush up on this, there are good engineering maths textbooks and resources on the Internet.  Having said that, we will give a very quick review of the meaning of the key terms used in the equations.

\oint_S {f\,da} - is the definite integral of function f(x,y,z), taken over a surface S

\oint_C {f\,dl} - is the line integral of function f(x,y,z), over a curve C

\vec \nabla \equiv \hat x\frac{\partial }{{\partial x}} + \hat y\frac{\partial }{{\partial y}} + \hat z\frac{\partial }{{\partial z}} - del (∇) performs derivative operations

\hat X, \vec X \cdot \vec Y and \vec X \times \vec Y - are the unit vector, vector dot product and vector cross product

Constants

{\varepsilon _0} - permittivity of free space (= 8.854187817... F/m)
{\mu _0} - permeability of free space (= 4π×10−7 ≈ 1.2566370614...×10−6 H/m)

Summary

This note has introduced the four Maxwell equations:

  1. Gauss's Electric Field Law - static electric fields
  2. Gauss's Magnetic Field Law - static magnetic fields
  3. Faraday's Law - electric fields induced by time varying magnetic fields
  4. Ampere Maxwell Law - magnetic fields induced by time varying electric fields

Each of these laws is explained in detail, in associated notes (listed above).   If you have any general comments, please discuss below.  If you have a detailed comment on a particular law, please discuss in the detailed note.  



Steven McFadyen's avatar Steven McFadyen

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

myElectrical Engineering

comments powered by Disqus

  1. Notes's avatar Notes says:
    9/4/2013 2:49 PM

    Trackback from Notes

    Gauss's Electrical law defines the relation between charge ("Positive" & "Negative") and electric field.  The law was initially formulated by Carl Friedrich Gauss in 1835. In Gauss's law, the electric field is the electrostatic field.  The law shows... ...


Comments are closed for this post:
  • have a question or need help, please use our Questions Section
  • spotted an error or have additional info that you think should be in this post, feel free to Contact Us



IEC 61439 Verification Methods

The (relatively new) switchgear and control gear standard, IEC 61439 'Low-voltage switchgear and controlgear assemblies' has three methods which can be...

Copyright Infringement

myElectrical does not support or promote the use of copyrighted material without the copyright owner's consent. If you believe that material for which...

8 Motor parts and common faults

Straight forward list of some common motor faults.  If I have missed any other common faults, please take a bit of time to add them in as a comment below...

Smarter Electrical Distribution

The other day I came across an article in Technology Review on the development of a smart transformer. A professor at North Carolina State University is...

Surface Treatment – Ladders, Trays and Baskets

Steel ladders, trays and baskets form the backbone of cable containment systems. Often these items need some form of surface treatment to prevent corrosion...

Cable Sheath and Armour Loss

When sizing cables, the heat generated  by losses within any sheath or armour need to be evaluated. When significant, it becomes a factor to be considered...

Software Usage Guidelines

Using software in our  work is essential for most of us and we are becoming even more dependant on it's use.  While software is a great asset, many times...

Maxwell's Equations - Introduction

Maxwell's Equations are a set of fundamental relationships, which govern how electric and magnetic fields interact. The equations explain how these fields...

What is LED?

Light Emitting Diodes (LED ) are increasing gaining favour in both the domestic and commercial sectors; due to their efficiency, sustainability and durability...

Magicians of Engineering

The other day I was reading 'Night of the New Magicians' by Mary Pope Osborn with my son.  The story is about a young boy and girl who travel back in time...

Have some knowledge to share

If you have some expert knowledge or experience, why not consider sharing this with our community.  

By writing an electrical note, you will be educating our users and at the same time promoting your expertise within the engineering community.

To get started and understand our policy, you can read our How to Write an Electrical Note