1.4. How to do better on exams[[Student version, December 8, 2002]] 19
product to beML^2 /T^2 .That’s anenergy.What sort of energy scales are relevant to our problem?
It occurs to you that the energy of thermal motion,Ethermal(to be discussed in Chapter 3), is
relevant to the physics of friction, since friction makes heat. So you could guess that if there is any
fundamental relation, it must have the form
ζD=?Ethermal. (1.10)Youwin. You have just guessed a true law of Nature, one that we will derive in Chapter 4. In
this case Albert Einstein got there ahead of you, but maybe next time you’ll have priority. As we’ll
see, Einstein had a specific goal: By measuring bothζandDexperimentally, he realized, one could
findEthermal.We’ll see how this gave Einstein a way to measure how big atoms are, without ever
needing to manipulate them individually. And...Atoms really are that size!
What did we really accomplish here? This isn’t the end, it’s the beginning: We didn’t find any
explanationof frictional drag, nor of diffusion, yet. But we know alotabout how that theory should
work. It has to give a relation that looks like Equation 1.10. This helps in figuring out the real
theory.
1.4.4 Some notational conventions involving flux and density
Toillustrate how units help us disentangle related concepts, consider a family of related quantities
that will be used throughout the book. (See Appendix A for a complete list of symbols used in the
book.)
- Wewill often use the symbolsNto denote the number of discrete things (a dimen-
sionless integer),Vto denote volume (with SI unitsm^3 ), andqto denote a quantity
of electric charge (with dimensionscoul). - The rates of change of these quantities will generally be written dN/dt(with units
s−^1 ),Q(the “volume flow rate,” with unitsm^3 s−^1 ), andI(theelectric current,
with unitscoul s−^1 ). - If we have five balls in a room of volume 1000m^3 ,wesaythat thenumber density(or
“concentration”) of balls in the room isc=0. 005 m−^3. Densities of dimensional
quantities are traditionally denoted by the symbolρ;asubscript will indicate what
sort of quantity. Thusmass densityisρm(unitskg m−^3 ), whilecharge densityis
ρq(unitscoul m−^3 ). - Similarly, if we have five checkers on a 1m^2 checkerboard, thesurface number density”
is 5m−^2 .Similarly, thesurface charge density”σqhas unitscoul m−^2. - Suppose we pour sugar down a funnel, and 40 000 grains fall each second through an
opening of area 1cm^2 .Wesaythat thenumber flux(or simply “flux”) of sugar
grains through the opening isj=(40 000s−^1 )/(10−^2 m)^2 =4· 108 m−^2 s−^1 .Similarly,
the fluxes of dimensional quantities are again indicated using subscripts; thusjqis
thecharge flux(with unitscoul m−^2 s−^1 )) and so on.
If you accidentally use number density in a formula requiring mass density, you’ll notice that your
answer’s units are missing a factor ofkg;this is your signal to go back and introduce the mass of
each object, convertingctoρm.