Week 10: Maxwell’s Equations and Light
I have also a paper afloat, with an electromagnetic theory of light, which, till I am con-
vinced to the contrary, I hold to be great guns.
James Clerk Maxwell (1831-1879) Scottish physicist. In a letter toC. H. Cay, 5
January 1865.
- Ampere’s Law has a bit of a problem. The currentthrough Cis not consistently
defined so that it gives the same value forallsurfacesSthat are bounded by the
closed curveC(through which we evaluate the flux of the current density to find
the current “throughC”). This means that two people can evaluate the integral to
find the current throughCand getdifferent answerswithout either of them making
a mistake. One can prove anything from a theory with an inconsistency, so this is
abad thing.
- James Clerk Maxwell noted this problem, and sat down toinventthe mathematical
tools and concepts to resolve it. We will proceed far more elegantly than he was able
to, using the gift of hindsight. Either way, we will all arrive at the followingconsis-
tentform for Ampere’s Law, one to which we have addedMaxwell’s Displacement
Current: ∮
C
B~·d~ℓ=μ 0
(∫
S/C
J~·nˆdA+d
dtǫ^0
∫
S/C
E~·nˆdA
)
Both of these latter two integrals must be evaluated with thesamesurfaceS, but
given this they sum together to give the same invariant current forallthe surfaces
Sthat are bounded by the closed curveC.
- In this new,correctversion of Ampere’s Law, you can see Maxwell’s contribution:
theMaxwell Displacement Currentproduced by atime varying electric field:
IMDC=
d
dt
ǫ 0
∫
S/C
E~·ˆndA
- It is worth writing down the complete set of trading cards, suitable for engraving:
∮
S
E~·ˆndA =^1
ǫ 0
∫
V/S
ρedV (775)
∮
S
B~·ˆndA = μ 0
∫
V/S
ρmdV = 0 (776)
∮
C
B~·d~ℓ = μ 0
(∫
S/C
J~·nˆdA+d
dt
ǫ 0
∫
S/C
E~·nˆdA
)
(777)
∮
C
E~·d~ℓ = −d
dt
∫
S/C
B~·ˆndA (778)
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