W9_parallel_resonance.eps

354 Week 10: Maxwell’s Equations and Light

Either Maxwell’s Equations arewrong, orclassical Newtonian mechanics itselfis wrong!

In which case everything we’ve learned over the last two semestersis wrong.

Too bad, ladies and gentlemen. Maxwell’s Equations appear to becorrect. Classical

Mechanics isnot. Visualize Newton, spiralling down to the earth in flames, never to rise

again as high as he was before Maxwell made this momentous discovery. Over the last

century it has been replaced byquantummechanics, awavetheory of matter that is,

to say the least, a lot more complicated than the relativelysimple F~=m~athat has

governed nearly everything so far.

The last thing I want to mention before we return to our regularly scheduled (but false)

“classical” treatment of electomagnetism is thatnearly all electromagnetic radiation

comes from oscillatingdipoles, predominantly electric dipoles at that. Our Lorentz model

atom turns out to be very useful (all the way into graduate classical mechanics) for

understanding the “generic” properties of radiating atoms and matter interacting with

a time dependent electromagnetic field.

Back to work.

10.1.2: The Wave Equation

B

z

x

y

∆y

∆x

∆z

Ex(z)

Ex(z + z)

By(z)

By(z +∆z)

∆

Figure 141: Two particular components of the electric and magneticfield, in a coordinate frame
“far” from any sources and varying in space and time. The graph is asnapshot at a particular time
t, but we can imagine thatEx(z, t) andBy(z, t) generally and ignore any other variation withxor
yfor the moment.

Let us start, then, withnosource terms in Maxwell’s equations, or rather, in a region of

space far from any sources. That doesn’t mean that the fields there are zero, only that

we don’t have to worry about how the fields were originally produced –we know that

they were somehow created by electric charges and currents butwe don’t care about the

details. Maxwell’s equations are then somewhat simpler:

∮

S

E~·ˆndA = 0 (843)

∮

S

B~·ˆndA = 0 (844)

∮

C

B~·d~ℓ = μ 0 ǫ 0 d

dt

∫

S/C

E~·ˆndA (845)

∮

C

E~·d~ℓ = −d

dt

∫

S/C

B~·nˆdA (846)

as now there are no magneticorelectric monopoles present, only the fields.