18 Chapter 1. What the ancients knew[[Student version, December 8, 2002]]
paragraph, this sum makes no sense unlessxis dimensionless. (Recall also that the sine function’s
argument is anangle,and angles are dimensionless.)
Suppose you run into a new constant in a formula. For example the force between two charged
objects in vacuum is
f=1
4 πε 0q 1 q 2
r^2. (1.8)
What are the dimensions of the constantε 0 ?Just compare:
MLT−^2 =[ε 0 ]−^1 Q^2 L−^2.In this formula the notation [ε 0 ]means “the dimensions ofε 0 ”; it’s some combination ofL,M,T,Q
that we want to find. Remember that numbers like 4πhave no dimensions. (After all,πis
the ratio of two lengths, the circumference and the diameter of a circle.) So right away we find
[ε 0 ]=Q^2 T^2 L−^3 M−^1 ,which you can then use to check formulas containingε 0.
Finally, dimensional analysis helps you remember things. Suppose you’re faced with an obscure
SI unit, like “farad” (abbreviatedF). You don’t remember its definition. You know it measures
capacitance, and you have some formula involving it, sayE=^12 q^2 /CwhereEis the stored electrical
energy,qis the stored charge, andCis the capacitance. Knowing the units of energy and charge
gives that thedimensionsofCare [C]=T^2 Q^2 M−^1 L−^2 .Substituting the SIunitssecond, coulomb,
kilogram, and meter, we find that the natural SI unit for capacitance iss^2 coul^2 kg−^1 m−^2 ). That’s
what a farad really is.
Example Appendix B lists the units of the permittivity of empty spaceε 0 asF/m. Check
this.
Solution: Youcould use Equation 1.8, but here’s another way. The electric potential
V(r)adistancerawayfrom a point chargeqisV(r)=
q
4 πε 0 r. (1.9)
The potential energy of another chargeqsitting atrequalsqV(r). Since we know
the dimensions of energy, charge, and distance, we work out [ε 0 ]=T^2 Q^2 M−^1 L−^3 ,
as already found above. Comparing what we just found for the dimensions of ca-
pacitance gives that [ε 0 ]=[C]/L,sothe SI units forε 0 are the same as those for
capacitance per length, orFm−^1.1.4.3 Using dimensional analysis to formulate hypotheses
Dimensional analysis has other uses. Let us see how it actually lets usguess new physical laws.
Chapter 4 will discuss the “viscous friction coefficient”ζfor an object immersed in a fluid. This
is the force applied to the object, divided by its resulting speed, so its dimensions areM/T.Wewill
also discuss another quantity, the “diffusion constant”Dof the same object, which has dimensions
L^2 /T.BothζandDdepend in very complicated ways on the temperature, the shape and size of
the object, and the nature of the fluid.
Suppose now that someone tells you that in spite of this great complexity, theproductζDis
very simple: This product depends only on the temperature, not on the nature of the object nor
even on the kind of fluid it’s in. What could the relation be? You work out the dimensions of the