8.10: Maxwell’s Equations- Summarizing the Connections Between Electricity and Magnetism

Learning Objectives

• Restate Maxwell’s equations.

The Scotsman James Clerk Maxwell (1831–1879) is regarded as the greatest theoretical physicist of the 19th century. (See Figure \(\PageIndex{1}\)). Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by Maxwell’s equations, he also developed the kinetic theory of gases and made significant contributions to the understanding of color vision and the nature of Saturn’s rings.

Figure \(\PageIndex{1}\): James Clerk Maxwell, a 19th-century physicist, developed a theory that explained the relationship between electricity and magnetism and correctly predicted that visible light is caused by electromagnetic waves. (credit: G. J. Stodart)

Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Let us first summarize some of the previous observations from this chapter so we can see their connections. Maxwell's equations are summarizing the results of many different types of work in electricity and magnetism and state the same general ideas as we have just summarized, but do so with mathematical rigor.

The first law we looked at for electricity was Coulomb's Law. This law was applied to point charges, but we could extend the concept to look at electric fields as well. This would result in a similar mathematical concept known as Gauss's Law. (Similar to what we saw with magnetism, it is difficult to express this law in equation form without getting into advanced mathematics, so we will not do so here.) As we extended our discussion of electricity into the idea of an electric field, we saw some characteristics of an electric field, namely that the fields originate at positive charges and move towards negative charges.

Figure \(\PageIndex{2}\): Characteristic behavior of electric fields. The field lines start on the positive charge and move towards the negative charge.

When we considered magnetic fields, we saw a somewhat different behavior. Instead of starting on a charge and extending to another charge, as we saw with electric fields, magnetic fields form loops that move through the magnet and back again. Whereas we can have discreet electrical charges, this was not observed for magnetic fields. In fact, one of the phenomena we introduced related to magnetism was that magnetic monopoles do not exist.

Figure \(\PageIndex{3}\): Characteristic behavior of magnetic fields. The field lines form loops and there are no monopoles.

The behavior of electricity and magnetism are distinctly different, but also mirrors of each other. Electrical fields extend out from electrical charges to other electrical charges. They have a beginning and an end. Magnetic fields do look similar to electric fields, but they do not extend from discreet points, but form loops. The lack of monopoles prevents the possibility of these discreet points from existing. The beauty of Maxwell's equations are the elegant way in which the mathematics shows these connected, but very distinct ideas.

Maxwell also incorporated the ways in which these fields interacted with each other into his analysis. Faraday had found that a changing magnetic field induces an electric field. And the work of Oersted and Ampere led to the analagous conclusion by Maxwell that moving charges result in magnetic fields. The two phenomenon, electricity and magnetism, were inextricably linked together regardless of how different they seemed to be. Maxwell's equations also linked these ideas together.

The results of Maxwell’s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts—why mathematics is the language of science.

THE RESULTS OF MAXWELL’S EQUATIONS

1. Electric field lines originate on positive charges and terminate on negative charges. The electric field is defined as the force per unit charge on a test charge.
2. Magnetic field lines are continuous, having no beginning or end. No magnetic monopoles are known to exist.
3. A changing magnetic field induces an electric field.
4. Magnetic fields are generated by moving charges or by changing electric fields.

Maxwell’s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday’s law of induction and had been suspected for some time, but fits beautifully into Maxwell’s equations. Maxwell's equations showed that electric and magnetic forces are not separate, but different manifestations of the same thing—the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature—the gravitational, electrical, strong, and weak nuclear forces.

Section Summary

• Maxwell’s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell’s equations.
• These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss’s law for electricity, second is Gauss’s law for magnetism, third is Faraday’s law of induction, including Lenz’s law, and fourth is Ampere’s law in a symmetric formulation that adds another source of magnetism—changing electric fields.

Glossary

Maxwell’s equations

a set of four equations that comprise a complete, overarching theory of electromagnetism

electric field lines

a pattern of imaginary lines that extend between an electric source and charged objects in the surrounding area, with arrows pointed away from positively charged objects and toward negatively charged objects. The more lines in the pattern, the stronger the electric field in that region

magnetic field lines

a pattern of continuous, imaginary lines that emerge from and enter into opposite magnetic poles. The density of the lines indicates the magnitude of the magnetic field