W9_parallel_resonance.eps

346 Week 10: Maxwell’s Equations and Light

Ampere’s Law and the Maxwell Displacement Current

As discussed at the end of week 8, Maxwell’s Equations – so far – don’tseem quite right.

Let’s write them out as we have them at this point:

∮

S

E~·nˆdA =^1

ǫ 0

∫

V/S

ρedV (807)

∮

S

B~·nˆdA = μ 0

∫

V/S

ρmdV= 0 (808)

∮

C

B~·d~ℓ = μ 0

∫

S/C

J~·nˆdA (809)

∮

C

E~·d~ℓ = −d

dt

∫

S/C

B~·nˆdA (810)

The asymmetry will be a bit more apparent if I put all of the terms involvingchargesas

sourcesof 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 (811)

∮

C

B~·d~ℓ = μ 0

∫

S/C

J~e·ˆndA (812)

∮

S

B~·nˆdA = 0 (813)

∮

C

E~·d~ℓ+d

dt

∫

S/C

B~·nˆdA = 0 (814)

I put a tinyesubscript on theJ~and reordered themwith a big hole in Ampere’s Lawto

emphasize the point. The top two equations are connected toelectrical charge– either

stationary or moving – to produce the fields. The bottom two are zero on the right, where

the zero just means “there ain’t no stinkin’ magnetic monopoles been seen (yet)” but we

canimaginethat if there were, Gauss’s Law for Magnetism would get a source term on

the right that looked just like that for Gauss’s Law for Electricity, and Faraday’s Law

would get a term on the right involving thecurrent densityofmovingmagnetic charge,

just like Ampere’s Law.

But what about poor Ampere’s Law, in that case? Faraday’s Law mixes electric and

magnetic fields, so that time varying magnetic fields make electric fields.

Shouldn’t Ampere’s Law have a term such that time varying electric fields make magnetic

fields? I left the gap just in case...

This is as good a thing as any to motivate a closer look at Ampere’s Law. Maxwell’s

Equations are starting to look ratherbeautiful^95 but that big hole isugly, as is (really)

the big ugly zeros where magnetic monopoles should live. Natural philosophers have

from time immemorial considered “beauty” – a certain appealing symmetry, as it were

• to be an essential component of probable truth. Sometimes this belief is followed to a
  fault, of course, especially when the beautiful idea in question isouridea, and ultimately
  nature itself is the arbiter of truth in natural law, but still, at the very least things that
  arealmostbeautifully symmetric demand a closer examination to see if we are missing
  something. Experimentalists today search for magnetic monopoles; we ourselves will
  follow in Maxwell’s footsteps and search for the missing term.

(^95) Seriously. If there is such a thing in this Universe as beautiful mathematics, Maxwell’s Equations are It. This
course won’t cover the half of just how gorgeous they really are...