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Physics

Chapter 22 (All)

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Maxwell's Laws for Electromagnetic Fields

Introduction

My theory of electrical forces is that they are called into play in insulating media by slight electric displacements, which put certain small portions of the medium into a state of distortion which, being resisted by the elasticity of the medium, produces an electromotive force ... I suppose the elasticity of the sphere to react on the electrical matter surrounding it, and press it downwards.
From the determination by Kohlrausch and Weber of the numerical relation between the statical and magnetic effects of electricity, I have determined the elasticity of the medium in air, and assuming that it is the same with the luminiferous ether I have determined the velocity of propagation of transverse vibrations.
The result is 193088 miles per second (deduced from electrical & magnetic experiments).
Fizeau has determined the velocity of light = 193118 miles per second by direct experiment.
This coincidence is not merely numerical. I worked out the formulae in the country, before seeing Webers [sic] number, which is in millimetres, and I think we have now strong reason to believe, whether my theory is a fact or not, that the luminiferous and the electromagnetic medium are one.

— James Clerk Maxwell, Letter to Michael Faraday (19 October 1861)

Outline

Maxwell's Laws

The development of electromagnetic field theory relied on the efforts of many people. Michael Faraday's work made it possible to control current flow in circuits, which opened up opportunities for more experimentation. The German Gustav Kirchhoff identified rules which allowed him to predict how different circuit configurations would affect voltage and current. He realized that electrical signals in wire propagate at the speed of light, even though individual electrons move at much slower speeds. This, with the information Faraday had already collected on polarized light in magnetic fields, gave James Clerk Maxwell some ideas about the relationship between electromagnetic phenomena and light.

Developing the Laws

Maxwell was a mathematician, trained at Edinburgh University in Scotland. He used mathematics to create models of physical systems and describe their limitations. One of his first analyses showed that Saturn's rings had to be made of small particles capable of individual movement at different speeds, because a solid object of the ring's size and shape would be torn apart by Saturn's gravitational field.

Aided by his wife, Maxwell performed a number of experiments on gas, color, and electromagnetic forces. He proposed that magnetism was the result of spinning molecules, which he called molecular vortices. The direction and speed of rotation of a vortex determined the intensity of its magnetic force. Using calculus to sum up the contributions of individual particles to an overall field, he was finally able to explain electromagnetic phenomena in precise mathematical terms.

Today, Maxwell's equations are considered the prime example of elegance, the physicist's term for a supremely simple mathematical relationship between basic natural concepts. All electromagnetic phenomena can be explained using one or more his his four equations, which are generalizations of the discoveries of his predecessors. These are

The Four Equations

The mathematical expression of Maxwell's laws rests on the calculus of vector fields. We've already met the fundamental vector operators, the dot product and cross product, used for single vectors. A vector dot product combines the magnitudes of two related vectors with by mapping one onto the other through the cosine of the angle between them. Work is a vector dot product between force and displacement: W = Fd = Fd cos θ. Torque is a vector cross product between the linear force and the radial vector to the axis of rotations: τ = F X r.

When we consider fields, we can look at the flow of individual vectors representing the field through a surface area, adding every vector at every point going out and subtracting every vector at every point pointing in. [The actual math requires integral calculus.] This is a kind of density measurement: the total number of vectors passing through a complete surface, divided by the volume of the enclosed surface. We represent this as the divergence of the field: ∇ • E.

Under the right conditions, a field can cause rotation in another object, even generating a new field as the thing rotating. In terms of physical objects, fluid flow can cause a ball floating the fluid to spin if its surface is rough enough. The direction of spin is, by convention, the axis pointing up through clockwise rotation, at right angles to the field and to the spin velocity. This new vector is the curl of the field, and is written as the cross product of the curl and the field: ∇ x E.

Name Simple representation Explanations
1: Gauss's Law ∇ • E   =   ρ ϵ 0 The electric flux through a surface bounding a charged volume depends on the density of the charge in the bounded volume. This is analogue to the idea that the amount of light flowing through the surface of a light bulb depends on the power of the light bulb.
2: Gauss's Law - magnetism ∇ • B   = 0 This would be the magnetic equivalent of electrical flux except that we never find an isolated magnetic pole. All lines out of the north pole are cancelled by lines into the south pole, so the divergence of the magnetic field is zero.
3: Faraday's Law of Induction ∇ × E   = δ B δ t A changing E field and its direction arising from a changing B field depends on the direction of the B field and its rate of change.
4: Ampere's Law ∇ × B   = μ 0 J +   μ 0 ϵ 0 δ E δ t A changing B field and its direction depends on the flow of current (if any) and any changing E field present (displacement current).

Notice that the last two laws reinforce each other: a changing magnetic field generates an electrical field, a changing electrical field generates a magnetic field, so the size or amplitude of the fields will oscillate (get bigger and smaller in a wave-like manner). A medium (the material through which the wave travels) can support the induction of magnetic and electrical fields in a specific amount, called the permittivity of the medium for electricity and the permeability of the medium for magnetism. The quantities of permittivity and permeability had been determined earlier for free space or vacuum. Maxwell realized that if he inserted them into his last two equations, he could predict the speed of the electromagnetic wave as 3.0 * 108 meters/sec. This coincided exactly for the independently—determined speed of light. Maxwell proclaimed in his Treatise on Electricity and Magnetism, published in 1873, that light was the result of electromagnetic wave propagation.

Practice with the Concepts

If the electric field in an EM wave has a peak magnitude of 0.72 * 10-4 V/m, what is the peak magnitude of the magnetic field strength?

The E field in an EM wave has a peak of 22.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time?

Discussion Points