Simple Nature - Light and Matter

a/James Clerk Maxwell (1831-
1879)

b/Where is the moving charge
responsible for this magnetic
field?

11.6 Maxwell’s equations

11.6.1 Induced magnetic fields

We are almost, but not quite, done figuring out the complete set

of physical laws, called Maxwell’s equations, governing electricity

and magnetism. We are only missing one more term. For clarity,

I’ll state Maxwell’s equations with the missing part included, and

then discuss the physical motivation and experimental evidence for

sticking it in:

Maxwell’s equations

For any closed surface, the fluxes through the surface are

ΦE= 4πkqin and

ΦB= 0.

For any surface that is not closed, the circulations around the edges

of the surface are given by

ΓE=−

∂ΦB

∂t

and

c^2 ΓB=

∂ΦE

∂t

+ 4πkIthrough.

The ΦEequation is Gauss’ law: charges make diverging electric

fields. The corresponding equation for ΦB tells us that magnetic

“charges” (monopoles) don’t exist, so magnetic fields never diverge.

The third equation says that changing magnetic fields induce curly

electric fields, whose curliness we can measure using the emf, ΓE,

around a closed loop. The final equation, for ΓB, is the only one

where anything new has been added. Without the new time deriva-

tive term, this equation would simply be Amp`ere’s law. (I’ve chosen

to move thec^2 over to the left because it simplifies the writing, and

also because it more clearly demonstrates the analogous roles played

by charges and currents in the ΦEand ΓBequations.)

This new∂ΦE/∂tterm says that just as a changing magnetic

field can induce a curly electric field, a changing electric field can

induce a curly magnetic field. Why should this be so? The following

examples show that Maxwell’s equations would not make sense in

general without it.

Figure b shows a mysterious curly magnetic field. Magnetic fields

are supposed to be made by moving charges, but there don’t seem

to be any moving charges in this landscape. Where are they? One

reasonable guess would be that they’re behind your head, where you

can’t see them. Suppose there’s a positively charged particle about

to hit you in the back of the head. This particle is like a current

722 Chapter 11 Electromagnetism