MOTION MOUNTAIN

liquid electricity, invisible fields and maximum speed 25
field is given by the local density of the field lines. The direction and the magnitude do
not depend on the observer. In short, the electric fieldE(x) is avector field. Experiments
show that it is best defined by the relation
qE(x)= dp(x)
dt
(2)
Challenge 7 e
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Challenge 8 s
Challenge 10 s
Ref. 8
taken at every point in spacex. The definition of the electric field is thus based on how
itmoves charges. In general, the electric field is a vector
E(x)=(E x ,E y ,E z ) (3)
and is measured in multiples of the unit N/C or V/m.
The definition of the electric field assumes that the test chargeqis so small that it does
not disturb the fieldE. We sweep this issue under the carpet for the time being. This is
a drastic move: we ignore quantum theory and all quantum effects in this way; we come
back to it below.
The definition of the electric field also assumes that space-timeis flat, and it ignores
all issuesdue tospace-timecurvature.
By the way, does the definition of electric field just given assume a charge speed that
is far less than that of light?
To describe the motion due to electricity completely, we need a relation explaining
how chargesproduce electric fields.This relation was established with precision (but not
for the first time) during the French Revolution by Charles-Augustin de Coulomb, on
his private estate.* He found that around any small-sized or any spherical chargeQat
rest thereisanelectricfield.Atapositionr,thiselectricfieldEisgiven by
E(r)= 1 Q r
4πε 0 r 2 r
where
1
4πε 0
=9.0 GV m/C . (4)
Later we will extend the relation for a charge in motion.The bizarre proportionality constant
is universally valid. The constant is defined with the so-called permittivity of free
spaceε 0 andisduetothehistoricalwaytheunitofchargewasdefinedfirst.**Theessential
point of the formula is the decreaseof thefield with thesquare of thedistance;can
youimaginetheoriginofthisdependence? AsimplewaytopictureCoulomb’sformula
isillustrated inFigure10.
The two previous equations allow us to write the interaction between two charged
bodies as
dp 1
dt
= 1 q 1 q 2 r
4πε 0 r 2 r =−dp 2
dt
, (5)
*Charles-AugustindeCoulomb(b.1736Angoulême,d.1806Paris),engineerandphysicist,provided,with
hiscareful experiments onelectriccharges,afirmbasisforthe studyofelectricity.
** Other definitions of this and other proportionality constants to be encountered later are possible,leading
to unit systems different from the SI system used here. The SI system is presented in detail in Appendix
A. Among the older competitors, the Gaussian unit system often used in theoretical calculations,
the Heaviside–Lorentz unit system, the electrostatic unit system and the electromagnetic unit system are
themostimportant ones.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net