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Maxwell’s Equations

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James Clerk Maxwell’s equations

In The Guardian, Alok Jha writes:

Maxwell’s Equations first appeared in “A dynamical theory of the electromagnetic field”, Philosophical Transactions of the Royal Society of London, in 1865. These are the equations of light, the mathematical relationships that showed us how to electrify our world and transmit energy and information through the air.

At the start of the 19th century, we lit our homes and offices with candles and oil lamps. Communications took the form of handwritten letters that took days to travel across the country, and several weeks across oceans. Today we use electricity to power everything and radio waves to talk to each other, anywhere around the world, instantaneously.

The seeds of that enormous change were planted in the 1830s, when the British physicist Michael Faraday built electric motors and showed that two natural forces, electricity and magnetism, were related. He proposed that these forces existed as “fields” that permeated space. In the latter half of the 19th century, the Scottish physicist James Clerk Maxwell formulated the equations that described these fields.

Maxwell modeled the fields as if they were invisible fluids that filled space. At each point in space, the electric field has a direction and a strength that can be measured if you put something there that can feel the effects of the field – an electron, say. If you could somehow measure the field at every point in space, you would know how it flowed and changed.

The two equations on the left in the picture show that the net flow of electric (E) and magnetic (H) field out of a closed volume of space, away from any electrical charges or magnetic materials, is zero.

The triangle and dot symbol in front of the field symbols (called the “divergence” operator) is a mathematical way to measure if a field behaves as a source or a sink at a specific point in space.

The equation for the magnetic field (H) stays the same even when there’s a magnet around – think of a bar magnet, the magnetic field lines around it start at the north pole and circle their way around to the south pole. And these field lines will always stop and start at a magnetic object, they do not appear or disappear in empty space.

The equation for the electrical field (E) is slightly different, though, when there are electrical charges around. A positive charge is a net source of electric fields and a negative charge is a sink. In that case, the net amount of field coming out of or into a volume is proportional to the charge contained within it.

The two equations on the right explain what happens when you move an electrical or magnetic field. The “curl” operator (the triangle and x symbol in combination) on the left of each equation is a way to measure a field moving in a tiny circle.

A changing electric field (E) produces a changing magnetic field (H). And vice versa.

The curly d/dts on the right measure a rate of change, a tiny change in a field (E or H) divided by a tiny change in time (t).

These equations are the basis of electromagnetic induction, the idea that if you move a magnet near an electrical conductor (or vice versa), you generate electricity. The electricity you use every day is made like this, using generators that work according to these equations.

Which leaves us with the letter “c” on the right side of the second two equations. This is a constant with a value of about 300,000 kilometres per second, which just happens to be the speed of light.

Bear in mind Maxwell did not put this in there because he was studying light; the number just popped out, unexpectedly, from the mathematics of the materials he was studying.

Maxwell had started by examining the properties of electricity and magnetism and stumbled upon a much deeper truth about them: the electromagnetic field was a medium for waves that, like ripples across the surface of a pond, travelled at a speed “c”. And the light we see is one of those electromagnetic waves.

That meant there should be other electromagnetic waves. The wavelengths we can detect with our eyes appear to us as colours. Shorter wavelengths of light include UV and gamma rays.

Longer wavelengths include heat (infra-red waves), microwaves and radio waves. The latter, of course, are the communications method of the 20th century – everything from radios to televisions to radar to mobile phones are based on our manipulation of the electromagnetic field, as described by Maxwell’s equations.

Maxwell’s equations

1. Gauss’ law:

Electric charge acts as sources or sinks for electric fields.

If you use the water analogy, + charge gives rise to flow out of a volume – this means + electric charge is like a source (a faucet – pumping water into a region).
Conversely, – charge gives rise to flow into a volume.
So – charge acts like a sink (fields flow into a region, and terminate on the charge).

This gives us a lot of intuition about the way fields can physically act in any scenario.
Here are possible and impossible situations for the Electric Field, as decided by the universe in the Law of Gauss it setup:


The D Field Must Have the Correct Divergence.

The Charges Dictate the Divergence of D .


2. Gauss’s law of magnetism

There are no magnetic monopoles; the total magnetic flux piercing a closed surface is zero.

In summary, the second of Maxwell’s Equations – Gauss’ Law For Magnetism – means that:

  • Magnetic Monopoles Do Not Exist
  • The Divergence of the B or H Fields is Always Zero Through Any Volume
  • Away from Magnetic Dipoles, Magnetic Fields flow in a closed loop. This is true even for plane waves, which just so happen to have an infinite radius loop.

3. Maxwell-Faraday’s law

The voltage accumulated around a closed circuit is proportional to the time rate of change of the magnetic flux it encloses.


See the great discussion of this law http://www.maxwells-equations.com/faraday/faradays-law.php

4. Maxwell–Ampère circuit law

Electric currents and changes in electric fields are proportional to the magnetic field circulating about the area they pierce.

See the full discussion of this law at Maxwells-equations.com Ampere’s law
(We also have a local archive of that page, Amperes-circuital-law

The following are consequences of this law:

  • A flowing electric current (J) gives rise to a Magnetic Field that circles the current
  • A time-changing Electric Flux Density (D) gives rise to a Magnetic Field that circles the D field

=== outline from Giancoli Physics ===
Chapter 22: Electromagnetic waves
22.1: Changing electric fields produce magnetic fields: Maxwell’s equations<
22.2: Production of electromagnetic waves
22.3: Light as an EM wave and the EM spectrum
22.4 Measuring the speed of light
22.5 Energy in EM waves
22.6 Momentum transfer and radiation pressure
22.7 Radio and Television as EM waves: Wireless communication

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

6.MS-PS4-1. Use diagrams of a simple wave to explain that (a) a wave has a repeating pattern with a specific amplitude, frequency, and wavelength, and (b) the amplitude of a wave is related to the energy of the wave.

HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium.

HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. Clarification Statements:
• Emphasis is on qualitative information and descriptions.
• Examples of technological devices could include solar cells capturing light and
converting it to electricity, medical imaging, and communications technology.

Massachusetts Science and Technology/Engineering Curriculum Framework (2006)

6. Electromagnetic Radiation Central Concept: Oscillating electric or magnetic fields can generate electromagnetic waves over a wide spectrum. 6.1 Recognize that electromagnetic waves are transverse waves and travel at the speed of light through a vacuum. 6.2 Describe the electromagnetic spectrum in terms of frequency and wavelength, and identify the locations of radio waves, microwaves, infrared radiation, visible light (red, orange, yellow, green, blue, indigo, and violet), ultraviolet rays, x-rays, and gamma rays on the spectrum.

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