MOTION MOUNTAIN

challenge hints and solutions 343
Obviously,thereisnowayto keep1 gof positivecharge together, astherepulsive forces among
thecharges would beeven larger.
Challenge 12, page 28: ToshowthefullequivalenceofCoulomb’sandGauss’s‘laws’,firstshow
that it holds for a single point charge. Then expand the result for more than one point charge.
ThatgivesGauss’s ‘law’ inintegralform,asgiven justbeforethischallenge.
To deduce the integral form of Gauss’s ‘law’ for a single point charge, one has to integrate
overtheclosedsurface.Theessentialpointhereistonotethattheintegrationcanbecarried out
foraninversesquaredependenceonly.Thisdependenceallowstransformingthescalarproduct
between the local field and the area element intoanormal product between the charge and the
solidangleΩ:
qdA cosθ
E dA=
4πε 0 r 2 = qdΩ . (109)
4πε 0
Incasethatthesurfaceisclosedtheintegrationisthenstraightforward.
Todeduce thedifferentialformof(thestatic)Gauss’s ‘law’,namely
∇E= ρ ε 0
, (110)
makeuseofthedefinitionofthechargedensityρandofthepurelymathematicalrelation
∮ E dA=∫ ∇E dV , (111)
closed surface enclosed volume
Thismathematicalrelation,validforanyvectorfieldE,iscalled Gauss’stheorem.Itsimplystates
thatthefluxisthevolumeintegralofthedivergence.
To deduce the full form of Gauss’s law, including the time-derivative of the magnetic field,
includerelativisticeffects bychangingviewpointtoamovingobserver.
Challenge 14, page 28: Unchargedbodiescanattracteachotheriftheyaremadeofchargedconstituentsneutralizingeachother,andifthechargesareconstrainedintheirmobility.Thecharge
fluctuationsthenleadtoattraction.Mostmoleculesinteractamongeachotherinthisway;such
forcesarealsoatthebasisofsurfacetensioninliquidsandthusofdropletformation.
Challenge 15, page 28: No;batteriesonlyseparatecharges andpumpthemaround.
Challenge 17, page 31: The ratioq/m of electrons and that of the free charges inside metals is
notexactlythesame.
Challenge 19, page 33: Findoutawaytotesttheissue,performtheexperiment,andpublishit!
Challenge 20, page 40: Ifyoucan,publishtheresult.Researchershavetriedtoputpeopleonthe
ocean during clouded days,have tried experiments in dark rooms,but nothinghas been found
so far. The experiences of people in magnetic resonance imaging equipment is inconclusive so
far.
Challenge 22, page 45: No.
Challenge 24, page 46: Thecorrect versionofAmpère’s ‘law’ is
∇×B− 1 c 2 ∂E
∂t =μ 0j (112)
whereastheexpressionmentionedinthetextmissestheterm ∂E
∂t .
Foranotherwaytostatethedifference,seeRichard P. Feynman,Robert B. Leighton
& Matthew Sands, The Feynman Lectures on Physics, volume II, Addison Wesley, p. 21-1,
1977.Theycanberead onlineforfree athttp://www.feynmanlectures.info.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net