MOTION MOUNTAIN

challenge hints and solutions 351
Challenge 161, page 151: Forsmallmirrorsorlenses,likethoseusedinmicroscopes,massproduction
is easier for lenses. In contrast, large mirrors are much easier and cheaper to fabricate
thanlargelenses,becausemirrorsuselessglass,arelighter,andallowchangingtheirshapewith
actuators.
Challenge 162, page 153: Syrup shows an even more beautiful effect in the following setting.
Take a long transparent tube closed at one end and fill it with syrup. Shine a red helium–neon
laser into the tube from the bottom. Then introduce a linear polarizer into the beam: the light
seeninthetubewill formaspiral.By rotatingthepolarizer youcanmake thespiraladvanceor
retract.Thiseffect,called theopticalactivity ofsugar,isduetotheabilityofsugartorotatelight
polarizationand to a special property of plants:they make only oneof thetwo mirror forms of
sugar.
Challenge 164, page 154: Therelation,theso-called ‘law’ ofrefractionis
c 1
c 2
= sinα 1
sinα 2
. (115)
The particular speed ratio between vacuum (or air, which is almost the same) and a material
givestheindexofrefractionnofthatmaterial:
n= c 1
c 0
= sinα 1
sinα 0
(116)
Manyincorrectly call the ‘law’ ofrefraction ‘Snell’slaw’,or ‘Descartes’ law’ even thoughmany
othersfounditbeforethem(andeventhoughthefamily nameis ‘Snel’).
Challenge 165, page 156: Thethinlens formula is
1
d o
+ 1 d i
= 1 f . (117)
It is valid for diverging and converging lenses, as long as their own thickness is negligible. The
strengthofalenscanthusbemeasuredwiththequantity1/f.Theunit1m −1 iscalledadiopter;it
isusedespecially forreading glasses.Converging lenseshavepositive,diverging lenses negative
values.
However,thethinlensformula isonlyanapproximation,andisneverusedinlensdesign.It
isarelicofoldtextbooks.ModernlensdesignersalwaysuseGaussianopticforcalculations.(See,
forexample,Francis A. Jenkins & Harvey E. White,FundamentalsofOptics,McGraw-
Hill,1957.)
Challenge 167, page 157: A light microscope is basically made of two converging lenses. One
lens – or lens system – produces an enlarged real image and the second one produces an enlarged
virtual image of the previous real image. Figure 175 also showsthat microscopes always
turn images upside down. Due to the wavelength of light, light microscopes have a maximum
resolutionof about1µm.Note thatthemagnificationof microscopesis unlimited; whatislimitedistheirresolution.Thisisexactlythesamebehaviourshownbydigitalimages.Theresolution
issimplythesizeofthesmallest possiblepixelthatmakes sense.
The microscopeseems to havebeen invented by Girolamo Fracastroin 1538.The firstviable
microscopes were built in the Netherlands around 1590. Progress in microscopes was so slow
because glass and lens production was extremely difficult at those times, especially for small
lenses. Therefore, David Brewster proposed in 1819 to build a microscope using the lens of a
fish eye; when this idea was realized with the lens of an eel, it resulted in a microscope with
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net