MOTION MOUNTAIN

liquid electricity, invisible fields and maximum speed 49
Vol. IV, page 47
Page 74
Ref. 26
Challenge Page28 84s
the Lagrangian is proportional to thesquare of the intensive quantity.The minus sign in
the expression is the same minus sign that appears also inc 2 t 2 −x 2 : it results from the
mixing of electric and magnetic fields that is due to boosts.
The Lagrangian density can be used to define the classical action of the electromagneticfield:
S=∫ ε 0
2 E2 − 1 B 2 dtdV . (19)
2μ 0
As usual, the action measures the change occurring in a system; it thus defines the
amountofchangethatoccurswhenanelectromagneticfieldmoves. (The expression for
the change, or action, of a moving light beam reduces to the product of its intensity and
total phase change.)The action of an electromagnetic field thus increases with its intensityandwithitsfrequency.Asusual,thisexpressionfortheactioncanbeusedtodescribe
themotionoftheelectromagneticfieldbyusingtheprincipleof leastaction.Indeed,the
principle implies the evolution equations of the electromagnetic field, which are called
Maxwell’sfieldequationsofelectrodynamics.This approach is the simplest way to deduce
them. We will discuss the field equations in detail shortly.
The second invariant of the electromagnetic field tensor,4EB=−c trF ∗ F, is a
pseudoscalar; it describes whether the angle between the electric and the magnetic field
is acute or obtuse for all observers.*
The uses of electromagnetic effects
The application of electromagnetic effects to daily life has changed the world. For example,
the installation of electric lighting in city streets has almost eliminated the previously
so common night assaults.These and all other electrical devices exploit the fact
that charges can flow in metals and, in particular, that electromagnetic energy can be
transformed
— into mechanical energy – as done in loudspeakers, motors and muscles;
— into light – as in lamps, lasers, glass fibres, glow worms, giant squids and various deep
ocean animals;
— into heat – as in electric ovens, blankets, tea pots and by electric eels to stun and kill
prey;
— into chemical effects – as in hydrolysis, battery charging, electroplating and the brain;
*Thereisinfactathird Lorentzinvariant, farlessknown.Itisspecifictotheelectromagnetic fieldandisa
combination ofthefieldanditsvector potential:
κ 3 = 1 2 A μA μ F ρν F νρ −2A ρ F ρν F νμ A μ
=(A⋅E) 2 +(A⋅B) 2 −|A×E| 2 −|A×B| 2 +4 φ c (A⋅E×B)−(φ c ) 2
(E 2 +B 2 ) . (20)
ThisexpressionisLorentz(but notgauge) invariant; knowing itcan helpclarifyunclear issues,suchasthe
lack of existence of waves in which the electric and magnetic fields are parallel. Indeed, for plane monochromatic
waves all three invariants vanish in the Lorentz gauge. Alsothe quantities∂ μ j μ ,j μ A μ –jbeing
theelectric current–and∂ μ A μ areLorentzinvariants.(Why?) Thelastone,theframeindependenceofthe
divergenceofthefour-potential, reflectstheinvariance ofgaugechoice.Thegaugeinwhichtheexpression
issettozeroiscalled theLorentz gauge.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net