W9_parallel_resonance.eps

Week 5: Resistance 169

We are now in a position to be able to derive a beautiful form for theLaw of Charge Conser-
vation. Consider a simple closed surfaceS(like the ones we considered for Gauss’s Law) located
anywhere in space. We already know that the closed surfaceSencloses some volumeV /S, and we
already know how to compute the total charge inside:

Qin S=

∫

V/S

ρdV (298)

or, the total charge insideSis the integral of the charge density inside.

If charge can never be created nor destroyed, the only way the total charge inV can change is
ifcharge moves across the surfaceS! Charge can flowinto the volume throughSoroutof the
volume throughS, or both at the same time, but ifSis impervious to charge (say it is a “perfect
insulator”), the charge inside can never change.

Quantitatively, then, the total current throughShas to equal the rate of change of the total
charge inside. All we have to do is assign a choice for the direction ofnˆ– into or out of the volume

• and write this in differential/integral form. Let’s choose “out” because it then is consistent with
  Gauss’s Law (which will prove strangely useful to us later on!):

Iout=

∮

S

J~·nˆdA=−d

dt

∫

V/S

ρdV=−

dQin S

dt

(299)

which we rearrange as: ∮

S

J~·nˆdA+d

dt

∫

V/S

ρedV= 0 (300)

This equation isvery important! It is, in fact, alaw of nature, based on substantial empirical
evidence. It is thelaw of charge conservationwritten in mathematical form. Basically, it says that
the amount of charge inside any volume bounded by a closed surfacecan only decrease (increase) if
charge flowsout (or in) through the surface!The net charge inside cannot just poof into or out of
existence, it has to get there by coming in from outside^51.

Advanced: Differential Form and Maxwell’s Equations

If/when you take a more advanced course in electromagnetism, one of the very first things you will do
is apply the divergence theorem to the Law of Charge Conservation, Gauss’s Law, and expressions
containing flux integrals in general and convert them to vector differential form. Treating the
divergence theorem and doing this algebra is beyond the scope of this course (although advanced
students may have done it in the starred homework problem in the Gauss’s Law chapter earlier and
can get the same result with the same procedure here) but we put down the result (only) here for
completeness and to make it easier to make the connection in a future course.

The law of charge conservation in differential form is:

∇~·J~+∂ρe

∂t

= 0. (301)

This ends up being much more convenient for doing the math associated with solving serious elec-
trodynamics problems. It is also has a critical invariance property when one learns about thefour-
dimensional geometryassociated with the theory of special relativity – basically charge is conserved
in all inertial reference frames even when relativity is taken into account.

(^51) There is another way charges can appear inside the box that doesn’t violate this law – theycanbe created
or destroyed apair at a timein such a way that thenetcharge of the pairs remains zero. This actually happens
in high energy quantum mechanical collisions – making it beyond the scope of this course – but the creation of a
positron-electron pair does not violatenetcharge conservation.