230 Week 7: Sources of the Magnetic Field
- Magnetic field lines must formclosed loops.
- As we’ll shortly see, those closed loops must be caused by something. That something is
moving charge passing through the loops, at least at first.
To repeat: Gauss’s Law with no monopoles is anempiricalrule, and lack of evidence isn’t
positive evidence of lack! We don’t know if there are, or are not, magnetic monopoles somewhere in
the Universe; we only know that we haven’t seen anyso farwhen we’ve looked for them quite hard.
At any moment, though, a reproducible experiment that observedthem couldchangeGauss’s Law
for magnetism (as well as other Maxwell equations) and we’d all haveto work a bit harder to learn
electrodynamics. But it would very definitely be worth it to be able to understand why charge is
quantized.
Magnetic Flux
Although (to be honest) there isn’t a lot of point to it, this work wouldnot be complete without
a mention of the definition of magnetic flux through a surfaceSand the SI units associated with
magnetic flux. Magnetic flux, it seems,getsits own units where electric flux does not, although we
haven’t missed the lack of units for electric flux and wouldn’t miss not knowing the units of magnetic
fluxunlessyou (sigh) read papers or other textbooks and they refer to them. Since some of you
might one day do just that, I suppose we should define them.
The definition of magnetic flux through a surfaceSis (as should already be clear):φm=∫
SB~·nˆdA (496)(where in context we might well omit themsubscript). Its SI units are calledWebers, where 1
Weber is one Volt-Second or one Joule/Ampere, as you prefer.
There. That’s done. You may now forget Webers, withoutreallyforgetting Webers, if you know
what I mean. We won’t use them, I’m certainly not going to ask you what they are on a test or
quiz, but you should probably know the True Fact that they are theSI units of magnetic flux so you
can understand what people are talking about who use them in a sentence like “Gosh, my magnetic
flux feels like it is up to 2. 17 × 10 −^7 Webers today, no wonder I feel terrible.”
Maybe Ishouldgive you an assignment on this: Go up to a non-physics-educated friend who
is very pretentious and puts on intellectual airs and ask him or her if he/she was worried about
the rapidly varying webers through the local power grid in the latestsolar storm. Enjoy his or her
profoundly pained or puzzled look for the precious moment that it lasts... oh yeah, did I do that?
Schooled!
Note well that while we won’t use theunitsbyname, it will turn out that thedefinitionof
magnetic flux will be very important to us, both in Gauss’s Law for Magnetism (where zero or not,
it is important in what it tells us) and in the yet-to-be-learnedFaraday’s Law, our final Maxwell
equation. Next week.
7.2: The Magnetic Field of a Point Charge
In the previous section we tried to generalize Gauss’s Law for Electrostatics into a Gauss’s Law
for Magnetostatics, where static magnetic fields could be createdby magnetic “charges” (magnetic
monopoles) much the same way that static electric fields are created by electric charges. Using our
imagination, we readily succeeded, but alas when we went out into theworld to search for magnetic