Maxwell's Equations

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Maxwell's Equations, formulated around 1861 by James Clerk Maxwell describe the interrelation between electric and magnetic fields.[1]They were a synthesis of what was known about electricity and magnetism, particularly building on the work of Michael Faraday, Charles-Augustin Coulomb, Andre-Marie Ampere, and others. These equations predicted the existence of Electromagnetic waves, giving them properties that were recognized to be properties of light, leading to the (correct) realization that light is an electromagnetic wave. Other forms of electromagnetic waves, such as radio waves, were not known at the time, but were subsequently demonstrated by Heinrich Hertz in 1888. These equations are considered to be among the most elegant edifices of mathematical physics.

Maxwell's equations serve many purposes and take many forms. On the one hand, they are used in the solution of actual real-world problems of electromagnetic fields and radiation. On the other hand, they are the subject of admiration for their elegance. There are many T-shirts, typically obtainable on college campuses, sporting various forms of these equations.

What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various \nabla \cdot \mathbf{E} and \nabla \times \mathbf{E} symbols appearing in some of the equations are the divergence and curl operators, respectively.

They are usually formulated as four equations (but later we will see some particularly elegant versions with only two), and the equations are usually expressed in differential form, that is, as Partial Differential Equations involving the divergence and curl operators. They can also be expressed with integrals. They are often expressed in terms of four vector fields: E, B, D, and H, though the simpler forms use only E and B.

Name differential form integral form
Coulomb's law of electrostatics, or Gauss's Law: \nabla \cdot \mathbf{D} = \rho \oint_S  \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V
Absence of magnetic monopoles: \nabla \cdot \mathbf{B} = 0 \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0
Faraday's Law of Induction: \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}  = -  \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}
Ampère's Law, or the Biot-Savart Law, plus displacement current: \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t} \oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
 \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}

In these, E denotes the electric field, B denotes the magnetic field, D denotes the electric displacement field, and H denotes the magnetic field strength or auxiliary field. J denotes the free current density, and ρ denotes the free electric charge density.

Contents

Integral Form

The integral forms can be seen to be equivalent to the differential forms through the use of the general Stoke's Theorem. The form known as Gauss's Theorem (k=3) takes care of the equations involving the divergence, and the form commonly known as just Stokes' Theorem (k=2) takes care of those involving the curl.

We will say nothing further about the equations in integral form. The differential versions are the "real" Maxwell equations.

What the Four Equations mean

Coulomb's Law

The first equation is just Coulomb's law of electrostatics, manipulated very elegantly (as usual) by Faraday and Gauss. Coulomb's law simply says that the electric force between two charged particles acts in the direction of the line between them, is repelling if they have like charges and attracting if unlike, is proportional to the product of the charges, and is inversely proportional to the square of the distance between them:

F = \frac{q_1 q_2}{4 \pi \epsilon\ d^2}

In this case, the constant defining the strength of the electric force is 4πε in the denominator. More about that presently.

In SI units the charges are measured in Coulombs, the force in Newtons, the distance in Meters, and the value of ε is 8.854 \times 10^{-12} Coulombs2 per Newton meter2, or Farads per meter.

Michael Faraday reformulated the electric and magnetic forces in terms of fields He said that what was really happening was that each charge was creating an electric field (called E) that acted on the other charge. The field created by the charge q1, as observed at distance d, is

E = \frac{q_1}{4 \pi \epsilon\ d^2}

and points directly outward from that charge, in all directions. The force felt by charge q2 is

F = q2E

Now consider a sphere of radius d with the charge at the center. If ρ is the charge density in Coulombs per cubic meter (Maxwell's equations are in terms of densities), the total charge in some volume is the integral, over that volume, of ρ.

So we have

q = \int_V \rho\, \mathrm{d}V

Now the field at the surface of the sphere is

\frac{q_1}{4 \pi \epsilon\ d^2}, or \frac{1}{4 \pi \epsilon\ d^2} \int_V \rho\, \mathrm{d}V

That field is directly outward, perpendicular to the sphere's surface, and is uniform over the surface. The integral of the field over the surface is d2 times that (the surface area of the sphere is d2; this is why we have the pesky factor of in various formulas; remember that d is the distance, and hence is the sphere's radius, not its diameter), so

\int_V \frac{\rho}{\epsilon}\, \mathrm{d}V = \oint_S  \mathbf{E} \cdot \mathrm{d}\mathbf{A}

But, by Gauss's Theorem,

\oint_S  \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V

So

\int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V = \int_V \frac{\rho}{\epsilon}\, \mathrm{d}V

Since this is true for any volume, we have

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}

Now D = \epsilon\ E in the straightforward case (more about that later), so

\nabla \cdot \mathbf{D} = \rho

Absence of Magnetic Monopoles

The second of the equations is just like the first, but for the magnetic field. The divergence of B must be the spatial density of magnetic monopoles. Since they have never been observed (though various Grand Unified Theories might allow for them), the value is zero.

This wasn't formulated initially in terms of monopoles, but was actually a statement that magnetic "lines of force" (the lines that intuitively describe the field) never end. They just circulate around various conductors carrying electric current. In contrast to this, lines of the electric field can be thought to "begin" and "end" on charged particles.

Faraday's Law

The third equation contains the result of Faraday's experiments with "electromagnetic induction"—a changing magnetic field creates an electric field, and that electric field circulates around the area experiencing the change in total magnetic flux. (Remember that the curl operator measures the extent to which a vector field runs in circles.) We won't go into the details of his experiments, except to note that he discovered that moving a coil of wire (a loop to pick up a circulating electric field) through a magnetic field (for example, by putting it on a shaft and turning it) led to the invention of electric generators, and hence made a major contribution to the industrialization of the world. Not bad for a theoretician.

The Unification of Light with the Theory of Electromagnetism

It is possible, by means of experiments with electrical charges and currents, to find such a figure c2 that has turned out to be the square of the velocity of propagation of electromagnetic effects. It could be demonstrated from static measurements of forces between two "unite" charges and between two "unit" currents that c ≈ 3.00 x 10 8ms-1. When Maxwell had performed this calculation with his equations for the first time, he could conclude that the system of electric and magnetic field will be propagated by this velocity and he realized the intriguing equivalence of this constant with the already experimentally determined speed of the light. Thus, he suggested that the light is the wave motion of the same medium that causes electric and magnetic phenomena. Electricity and magnetism had been unified by experimental works of Faraday, Ørsted and Ampère. After them, Maxwell carried out one of the greatest unifications in the physical sciences, the light ceased to be "something different" and have become electricity and magnetism in a new form - electromagnetic field propagated across the space.[2]

Other Formulations

There are even more esoteric formulations (leading to more esoteric T-shirts) involving advanced concepts of differential geometry. See [1].

References

  1. Wile, Dr. Jay L. Exploring Creation With Physical Science. Apologia Educational Ministries, Inc. 1999, 2000
  2. Richard Phillips Feynman, Robert B. Leighton, Matthew Linzee Sands (1966). The Feynman Lectures on Physics, Volume 2, 2, Addison-Wesley, 18-9. 

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