W9_parallel_resonance.eps

352 Week 10: Maxwell’s Equations and Light

(in the right handed direction around the current onto the disk as shown).

If we choose the larger Amperian pathCatr > R, the only thing that changes is that

the flux is no longer a function ofr, as the field is nonzero only in between the plates

and equalsφC=Qǫ 0 there. The field (after the same basic algebra) becomes:

Bφ=

μ 0 I

2 πr

r > R (834)

Note two things. First, the two algebraic forms forBφare equal atr=R, the boundary

between the two regions. Second, on theinsidethe field is the same as the field one

would expect in a wire of radiusRcarrying a uniform currentI(and vanishes atr= 0

as might be expected), while on theoutsidethe field is that of an infinitely long straight

wire. These two observations are strong algebraic evidence that our displacement current

has indeed “solved” the problem of finding an invariant current thatgives us sensible

answers regardless of the pathCor surfaceSchosen that is bounded by it.

10.1: Maxwell’s Equations for the Electromagnetic Field: The Wave Equation

OK, so let’s rewrite the complete set of Maxwell’s Equations, but this timewithMaxwell’s

teensy weensy little contribution and see if we can figure out why it is so all-fired impor-

tant that physicists speak in hushed tones when they mention Maxwell’s name, much as

they do for Newton and Einstein and a handful of others:

∮

S

E~·ˆndA =^1

ǫ 0

∫

V/S

ρedV (835)

∮

S

B~·ˆndA = μ 0

∫

V/S

ρmdV= 0 (836)

∮

C

B~·d~ℓ = μ 0

(∫

S/C

J~·nˆdA+ǫ 0 d

dt

∫

S/C

E~·nˆdA

)

(837)

∮

C

E~·d~ℓ = −d

dt

∫

S/C

B~·ˆndA (838)

Thesymmetrywill now be a apparent if I put all of the terms involvingchargesassources

of the fields on the right and all of the terms involving the fields themselves on the left:

∮

S

E~·nˆdA =^1

ǫ 0

∫

V/S

ρedV (839)

∮

C

B~·d~ℓ−μ 0 ǫ 0 d

dt

∫

S/C

E~·nˆdA = μ 0

∫

S/C

J~e·nˆdA (840)

∮

S

B~·nˆdA = 0 (841)

∮

C

E~·d~ℓ+d

dt

∫

S/C

B~·nˆdA = 0 (842)

Theonlyasymmetry now arises from the empirical non-observation of magnetic monopoles,

and even you, humble beginning physics student that you are, can already see exactly

what we would have to do to “fix” Maxwell’s Equations if tomorrow somebody performed

a reproducible experiment that discovered them.

But this symmetry isn’t (yet) why Maxwell is cool. No, there is something much more

profound buried in these equations now. Faraday’s Law already showed us that changing