W9_parallel_resonance.eps

228 Week 7: Sources of the Magnetic Field

• The field of a long straight wire carrying a currentIis:

B~=^2 kmI

r

φˆ (488)

whereφˆcurls around the wire in the direction given by theright hand rule.

• Learn to use the Biot-Savart law to find the field of a long straight wire, a current carrying
  loop, and a rotating disk of charge. From either of the latter two (far from the disk or ring)
  you should be able to guess thegeneralmagnetic field of a magnetic dipole in terms of its
  dipole moment in analogy with the field of an electric dipole. (See homework)


• With more work than we can do in this course the Biot-Savart Law canbe used to prove
  Ampere’s Law: ∮

C

B~·d~ℓ=μ 0 Ithru C=μ 0

∫

S/C

J~·nˆdA (489)

This is ourthirdMaxwell equation.

• There is aconceptual errorin Ampere’s Law. The currentI through an open surfaceS
  bounded by a closed curveCisnot invariantas we vary all possible such surfaces! From this
  one observation, plus your knowledge thatcharge is conserved(so that the net flow of charge
  out of any closed volume must equal the rate at which the charge inside that volume decreases
  in time:
  dQ
  dt

=−

∮

S

J~·nˆdA (490)

you should be able todeducethe necessity forMaxwell’s Displacement Current(which makes

the total current invariant). If you can do this on your own without looking and show me the

algebra, you get a piece of candy! Sorry, you’re just a bit late for aNobel prize, but this is

the general idea for how you will eventually go about winning one. Findan inconsistency and

solve it. Unify a field. You too can have your name on something!

• Learn to use Ampere’s Law to find the magnetic field of any cylindricallysymmetric current
  distribution, a (long) solenoid, and a toroidal solenoid. (See homework)


• Useful true fact: We do not usually deduce ascalarmagnetic potential analogous to the
  electric potential. Instead you will eventually learn about avectorpotential that leads to the
  magnetic field by virtue of differentiation (the curl). Because it is a vector, it is not much
  easier to evaluate directly than the Biot-Savart law above (it involves doing a very similar but
  slightly simpler integral). We will therefore skip it altogether in this course.

7.1: Gauss’s Law for Magnetism

At this point we know a rather lot about the magnetic field. We know thatmoving chargesexperience
a magnetic force when they move through a magnetic field, and we further know that that force is
“odd” compared at least to the Coulomb electrostatic force which (like gravity) acted on the “right
line” connecting two charges. It is time to search for thesourcesof this field.

It is perfectly reasonable to begin our search by saying to ourselves: “Gee, I just spent all of
this time learning about electrostatic fields coupled to monopolar electrical charges that behave
likekeqe/r^2. I know about the gravitational field too, which behaves likeGm/r^2. Is it just barely
possible that there is a quantity that behaves like a gravitational mass or an electrical monopolar
charge that is similarly a source of the magnetic field?”