space,^2 while charge doesn’t, we can anticipate that the mathemat-
ical rule connecting a magnetic field to its source-currents will have
to be completely different from the one relating the electric field to
its source-charges. However, if you look carefully at the argument
leading to the relation dUe/dv = E^2 / 8 πk, you’ll see that these
mathematical details were only necessary to the part of the argu-
ment in which we fixed the constant of proportionality. To establish
dUe/dv∝E^2 , we only had to use three simple facts:- The field is proportional to the source.
- Forces are proportional to fields.
- Field contributed by multiple sources add like vectors.
All three of these statements are true for the magnetic field as well,
so without knowing anything more specific about magnetic fields —
not even what units are used to measure them! — we can state
with certainty that the energy density in the magnetic field is pro-
portional to the square of the magnitude of the magnetic field. The
constant of proportionality is given on p. 691.(^2) Current is a scalar, since the definitionI= dq/dtis the derivative of a
scalar. However, there is a closely related quantity called the currentdensity,J,
which is a vector, andJis in fact the more fundamentally important quantity.
610 Chapter 10 Fields