W9_parallel_resonance.eps

Week 7: Sources of the Magnetic Field

(Est 2/18-2/25)

• Noisolated magnetic monopoleshave been experimentally observed, in spite of an electromag-
  netic theory that “begs” for them, a quantum theory that can explain charge quantization if
  asinglemagnetic monopole exists in the Universe, in spite of an intense experimental search
  for them. It is probably safe to say that magnetic monopoles are atthe very leastrare.


• We express this (lack of monopoles) by means ofGauss’s Law for Magnetism:
  ∮

S

B~·nˆdA= 4πkmQm,in S=μ 0

∫

V/S

ρmdV= 0 (485)

where the magnetic field constantkm= 10−^7 tesla-meter/ampereexactly(exactly because it

defines the coulomb, not the other way around).

• The actual source for magnetic fields (in the absence of monopoles) ismoving charge. The
  field produced by a point charge is given by:

B~=kmq~v׈r

r^2

=

μ 0

4 π

q~v׈r

r^2

(486)

whereμ 0 = 4π× 10 −^7 tesla-meter/ampere is called themagnetic permeability of free spaceand

is the magnetic constant analoguous toǫ 0 , the dielectric permittivity of free space.

• If we consider a wire carrying a currentI=nqvdA(where recallvdis the average drift speed
  of the charge carriersq), the amount of charge in a small length of wiredℓisdq=nqAdℓ. The
  field it produces is therefore:

dB~ = km

dq~vd׈r

r^2

dB~ = km

nqAdℓ~vd×rˆ

r^2

dB~ = km

nqvdAd~ℓ׈r

r^2

dB~ = kmId

~ℓ×rˆ

r^2

whered~ℓis a differential length of the wire with a direction pointingin the direction of the

current. This:

dB~=kmId

~ℓ×rˆ

r^2

(487)

is known as theBiot-Savart Lawfor the magnetic field, and (one way or another) is the way

most of the magnetostatic fields we observe in nature come into being.

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