Essay 6Vector Fields andMaxwell’s EquationsWith the introduction of the magnetic field B , wenow have examples of three different vector fields.We started with the velocity field v representing theflow of an incompressible fluid like water. Then wehad the electric field E created by point charges likeprotons and electrons. And now the magnetic fieldwhich can be created by the flow of electrons in awire.

For the electric field of point charges, we noticedthat there was a mathematical analogy between theelectric field and the velocity field of an incompress-ible fluid. Thus we could use Gauss’ law and theconcept of flux tubes to introduce electric field lines.Gauss’ law gives us the simple rule that Q/ε0 fluxtubes or field lines start from a positive charge +Q orstop on a negative charge –Q. Field lines never crossor start or stop in the space between the charges.These simple rules plus an intuitive feeling forsymmetry allowed us to sketch field shapes likeFigure (19-15) and even solve problems like calcu-lating the electric field of a line of charge.

For the magnetic field we used iron filings or com-pass needles to show us the shape of the magneticfield produced by electric currents and iron magnets.In one case, where we had a current i in a straightwire, we actually calculated the strength of themagnetic field that was going in circles about thewire. The result was

B =

μ0i2πr (23-18)

We obtained this result, not from a general rule forcalculating magnetic fields, but instead fromCoulomb’s law and our knowledge of Einstein’sspecial theory of relativity.

There is also a general rule for calculating magneticfields, a rule somewhat analogous to Gauss’ law forelectric fields. The rule is known as Ampere’s lawand is discussed in the Satellite Chapter 10.

Q/ε = –3οQ/ε = 5ο

Figure 19-15Mapping electric field lines.

i

B

Figure 23-14Magnetic field of a straight current.

ESSAY 6 VECTOR FIELDS AND MAXWELL’SEQUATIONS

Follows Chapter 23

Essay 6-2 Vector Fields and Maxwell’s Equations

AMPERE’S LAW We will give a brief example of Ampere’s law as itapplies to the magnetic field of a straight wire, butwe will not attempt to use it here to solve problems.

To introduce Ampere’s law, consider the magneticfield line B going in a circle of radius r about acurrent i as shown in Figure (1). Writing Equation(23-18) in the form

B × 2πr = μ0i (1)

we see that if we multiply the magnitude of B timesthe circumference of the circle 2πr , the result isequal to a constant μ0 times the current i flowingthrough the circle.

Ampere’s law is a generalization of this result. Ittells you to walk around a complete path, comingback to the point where you started. The circle inFigure (1) is such a path. With each step you takewhile walking, multiply the length of your step bythe strength of the magnetic field in the direction ofyour step, and record the result. When you get backto your starting point and add up all your results, thesum will be the constant μ0 times the total amountof current i flowing through your path. In the case ofFigure (1), B is always pointing in the direction ofthe circular path, so that all we have to do is multiplyB times the total length of the path 2πr , to get μ0i .

In Satellite Chapter 10 Vector fields and Ampere’sLaw, we do a more detailed introduction to Ampere’slaw, and then use the law for calculating othermagnetic field structures, like the field inside a coilof wire.

TWO KINDS OF VECTOR FIELDSThe electric field E of a point charge, and themagnetic field B of a straight current, compared inFigure (2), are examples of two different kinds ofvector fields. The electric field of a point charge isa clear example of what we call a diverging kind ofvector field. The field lines diverge, or come straightout of the point charge.

The magnetic field surrounding a wire is an exampleof what we call a circulating vector field. Here,instead of diverging out of its source, it circulatesaround its source. One of the accomplishments ofthe mathematics of vector fields is to show that thesetwo kinds of fields have completely distinct sources.For example, electric charges at rest produce onlydiverging electric fields, and electric currents pro-duce only circulating magnetic fields.

B

r

i up

Figure 1Magnetic field of a current.

Q E

iup

B

Figure 2Two kinds of vector fields.

diverging

circulating

Vector Fields and Maxwell’s Equations Essay 6-3

As a test of this idea, we can ask what kind of a vectorfield is the velocity field v of an incompressiblefluid like water? Near the beginning of Chapter 18on fluids we discussed the velocity field of a pointsource of water, seen in Figure (18-6). In that figurewe imagined that water molecules were being cre-ated inside a small magic sphere. Beyond thatsphere there was no more magic, and we used thecontinuity equation to show that the velocity fielddropped off as 1/r2 as the fluid flowed out throughlarger spheres whose area increased as r2 .

The problem with that example is that for an incom-pressible fluid, there are no magic spheres. We donot see water actually diverging outward as shownin Figure (18-6). Instead we see flow patterns likethe bathtub vortex of Figures (18-25) and (18-26) orHurricane Allen in Figure (18-27). Because thereare no magic spheres creating water molecules, theonly kind of velocity field we see in an incompress-ible fluid is a circulating velocity field v . You willnot see the diverging velocity field of Figure (18-6).

r1

r2

v1

2v

2v2v

2v

2v

v1

v1v1

v1

point sourceof water

Figure 18-6Water molecules are created inside the smallmagic sphere. This creates a divergingvelocity field out from the small sphere.

core

Figure 18-26Velocity field of the bathtub vortex.

Figure 18-27Eye of hurricane Allen viewed from a satellite.

Figure 18-25Bathtub vortex in a funnel. We stirredthe water before letting it drain out.

Essay 6-4 Vector Fields and Maxwell’s Equations

ELECTROMAGNETIC FIELDSWe will now focus our attention on the two vectorfields involved in electricity and magnetism, E and

B . Because each of these fields could, at least inprinciple, have two components, a diverging kindand a circulating kind, there are four possible elec-tric and magnetic field types indicated in Figure (3).There is the diverging electric and circulating mag-netic fields we saw in Figures (3a) and (3b). Theother two possibilities are a circulating electric anda diverging magnetic field drawn in Figures (3c) and(3d).

It turns out that each kind of field has its own specialkind of source and its own special law for calculatingthe field produced by that source. For the divergingelectric field, the source is electric charge, and thelaw determining the field produced is Gauss’ law.For the circulating magnetic field, the source iselectric current and the law used to calculate theresulting magnetic field is Ampere’s law.

For the diverging magnetic field shown in (3d), thesource is the so called magnetic monopole. Sometheories of the early universe predict that magneticmonopoles should have been created shortly afterthe big bang. Physicists have spent years looking fora magnetic monopole, but so far have found none.Until they do find one, we have a very simple rulerule for calculating the diverging kind of magneticfield—there is none!

It may be a surprise, but the circulating kind ofelectric field not only exists, it plays a very impor-tant role in modern life. The source of a circulatingelectric field is a changing magnetic field. Thepractical applications which we could not live with-out is the electric generator. By changing themagnetic field passing through a wire loop youcreate an electric voltage around the loop, a voltagethat can be used by a power company to provideelectric voltage to your house.

The law we use to calculate the circulating kind ofelectric field is Faraday’s law discussed in theSatellite Chapter 11. Of all the satellite chapters, thismay be the one that should be included as part of theregular course, if time permits. In that chapter wediscuss and carry out an important experiment thathelped lead Einstein to the special theory of relativ-ity.

Q E

iup

B

E

B

3a) diverging electric field 3c) circulating electric field

3d) diverging magnetic field3b) circulating magnetic field

Figure 3Four possible kinds of electric and magnetic fields.

Satellite Chapter 10Vector Fieldsand Ampere’s Law

Satellite Chapter 11Faraday’s LawAND Generators

Satellite Chapter 12Magnetic Forceon an Electric Current

Related satellite chapters.

Vector Fields and Maxwell’s Equations Essay 6-5

MAXWELL’S EQUATIONSThe four equations governing the behavior of thefour components of electric and magnetic fields areknown as Maxwell’s equations. They are Gauss’law for diverging electric fields, Ampere’s law forcirculating magnetic fields, Faraday’s law for cir-culating electric fields, and the absence of magneticmonopoles for the absence of divergent magneticfields. Then why, with Gauss, Ampere and Faradayall preceeding Maxwell, do we call these Maxwell’sequations?

The answer begins with the fact that Maxwell dis-covered another source for the magnetic field. Wementioned that a changing magnetic field couldcreate a circulating electric field. Maxwell discov-ered that a changing electric field would act as asource of a circulating magnetic field.

At this point Maxwell saw a new possibility. Out inempty space where there are no charges or currents,he could still create electric and magnetic fields. Hehad an equation for how a changing magnetic fieldcould create an electric field. The electric field beingcreated this way could in turn create a magneticfield. The magnetic field being created this waycould create an electric field . . . etc. In other words,electric and magnetic fields could feed off of eachother and travel in a wavelike manner through emptyspace. He calculated the speed of this electro-magnetic wave and the result was the speed c givenby

c = 1μ0ε0

= 3 × 108meterssec

which is the speed of light.

As a result, Maxwell saw that he had discoveredwhat a light wave was—a wave of electric andmagnetic fields feeding off of each other, travelingas a wave through empty space. In the next chapterwe study the structure of this electromagnetic wave.

Essay 6-6 Vector Fields and Maxwell’s Equations