MOTION MOUNTAIN

challenge hints and solutions 347
Vol. V, page 233
making up the chromosomesand many other molecules in living systems, the right hand preference
of most humans, the asymmetry of fish species which usually stay flat on the bottom of
theseas.
Challenge 93, page 87: Thisisnotpossibleatallusinggravitationalorelectromagnetic systems
or effects. The only way is to use the weak nuclear interaction, as shown in the chapter on the
nucleus.
Challenge 94, page 87: The Lagrangian does not change if one of the three coordinates is
changedbyitsnegativevalue.
Challenge 95, page 88: Theimageflipsup:a90degree rotationturns theimageby180degrees.
Challenge 96, page 88: ImagineEandBastheunitevectorsoftwoaxesincomplexspace.Then
anyrotationoftheseaxesisalsoageneralized duality symmetry.
Challenge 97, page 91: Theangular momentumwasputintothesystemwhenitwasformed. If
we bring a point charge from infinity along a straight line to its final position close to a magnetic
dipole, the magnetic force acting on the charge is not directed along the line of motion.
It therefore creates a non-vanishing torque about the origin. See J.M. Aguirregabiria &
A. Hernandez, The Feynman paradox revisited, European Journal of Physics 2, pp. 168–170,
1981.
Challenge 98, page 92: Show that even though the radial magnetic field of a spherical wave is
vanishing by definition, Maxwell’s equations would require it to be different from zero. Since
electromagneticwavesaretransversal, itisalsosufficienttoshowthatitisimpossibletocomba
hairy spherewithouthaving a(double) vortex ortwo simplevortices. Despite thesestatements,
quantum theory changes the picture somewhat: the emission probability of a photon from an
excited atominadegeneratestateisspherically symmetric exactly.
Challenge 99, page 92: If the conservation of linear and angular momentum are taken intoaccount,thereisnoambiguityofthePoyntingvector.See,forexample,W.H.
Furry,Examplesof
momentum distributions in the electromagnetic field and inmatter, American Journal of Physics
37,pp.621–636,1969.
Challenge 100, page 92: Theemitted radiationisstronglysuppressedbecausethesizeofthedipole(theplug)ismuchsmaller
thanthewavelength ofthefield.
Challenge 102, page 92: No. Neither electromagnetic motors nor coils have been found in any
living system. Muscles, the most powerful actuators in biology, are mainly made of large numbers
of electrostatic motors. The fundamental reason for this difference is the low efficiency of
microscopicelectromagneticmotors,whichcontrastswiththehighefficiencyofmicroscopicelectrostaticmotors.Atmacroscopicsizes,theefficiency
advantagesswitches.
Challenge 104, page 99: Inevery caseofinterference,theenergy isredistributed intootherdirections.Thisisthegeneralrule;
sometimesitisquitetrickytodiscoverthisotherdirection.
Challenge 105, page 99: The author regularly sees about 7 lines; assuming that the distance is
around20μm, thismakes about3μm per line.The wavelength must besmaller than this value
and the frequency thus larger than100 THz. The actual values for various colours are given in
thetableoftheelectromagnetic spectrum.
Challenge 107, page 100: Thedistancelbetween thelinesofaninterference pattern isgiven by
l=λd/s,wheredisthedistancetothescreen,andsisthesourceseparation.
To learn more about interference and theconditionsfor its appearance, explore the concept
ofFresnelnumber.Forexample,theFresnelnumberallowstodistinguishthe‘farfield’ fromthe
‘nearfield’,twosituationsthatoccurinmanywavephenomena.
Challenge 108, page 101: He noted that when a prism produces a rainbow, a thermometer
placedintheregionafterthecolourred showsatemperaturerise.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net