6. The Logarithms Of Physics

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Snippets of Physics 6. The Logarithms of Physics T Padmanabhan

S c a lin g a rg u m e n ts a n d d im e n sio n a l a n a ly sis a re p o w e rfu l to o ls in p h y sic s w h ich h e lp y o u to so lv e se v e ra l in te r e stin g p ro b le m s. A n d w h e n th e sc a lin g a rg u m e n ts fa il, a s in th e e x a m p le s d isc u sse d h e r e , w e a re le d to a m o re fa sc in a tin g situ a tio n . T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.

L et u s b eg in th is tim e b y rev isitin g a p ro b lem w h ich is b ea ten to d ea th in sta n d a rd tex tb o o k s in electro d y n a m ics { ex cep t th a t w e w ill d o it in a slig h tly d i® eren t m a n n er a n d g et o u rselv es a ll tied u p in k n o ts. C o n sid er a n in ¯ n ite stra ig h t lin e ch a rg e lo ca ted a lo n g th e y ¡ a x is w ith th e ch a rg e d en sity p er u n it len g th b ein g ¸ . W e a re in terested in d eterm in in g th e electric ¯ eld ev ery w h ere d u e to th is lin e ch a rg e. T h e sta n d a rd so lu tio n to th is p ro b lem is rid icu lo u sly sim p le. Y o u ¯ rst a rg u e, b a sed o n th e sy m m etry, th a t th e electric ¯ eld a t a n y g iv en p o in t is in th e x ¡ z p la n e a n d d ep en d s o n ly o n th e d ista n ce fro m th e lin e ch a rg e. S o w e ca n a rra n g e th e co o rd in a te sy stem su ch th a t th e p o in t a t w h ich w e w a n t to ca lcu la te th e ¯ eld is a t (x ;0 ;0 ). If w e n ow en clo se th e lin e ch a rg e b y a n im a g in a ry co n cen tric cy lin d rica l su rfa ce o f ra d iu s x a n d len g th L , th e o u tw a rd ° u x o f electric ¯ eld th ro u g h th e su rfa ce is 2 ¼ x L E w h ich sh o u ld b e eq u a l to 4 ¼ tim es th e ch a rg e en clo sed b y th e cy lin d er, w h ich is 4 ¼ L ¸ . T h is im m ed ia tely g iv es E = (2 ¸ = x ). [Y o u w o u ld h av e n o ticed to y o u r su rp rise th a t I a m u sin g th e cg s u n its; th e S I p eo p le sh o u ld rep la ce 4 ¼ b y (1 = ² 0 ).] D im en sio n a lly, electric ¯ eld is ch a rg e d iv id ed b y sq u a re o f th e len g th a n d sin ce ¸ is ch a rg e p er u n it len g th , ev ery th in g is ¯ n e.

Keywords Logarithm, potential theory.

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W e w ill n ow d o it d i® eren tly a n d in { w h a t sh o u ld b e { a n eq u iva len t w ay. W e w a n t to co m p u te th e electro sta tic p o ten tia l Á a t (x ;0 ;0 ) d u e to th e lin e ch a rg e a lo n g th e y ¡ a x is a n d o b ta in th e electric ¯ eld b y d i® eren tia tin g . O b v io u sly, th e p o ten tia l Á (x ) ca n o n ly d ep en d o n x a n d ¸ a n d m u st h av e th e d im en sio n o f ch a rg e p er u n it len g th . If w e ta k e Á » ¸ n x m , d im en sio n a l a n a ly sis im m ed ia tely g iv es n = 1 a n d m = 0 , so th a t Á (x ) / ¸ a n d is in d ep en d en t o f x ! T h e p o ten tia l is a co n sta n t a n d th e electric ¯ eld va n ish es! W e a re in tro u b le. A n ex p licit co m p u ta tio n o f th e p o ten tia l fro m ¯ rst p rin cip les m a k es m a tters w o rse. A n in ¯ n itesim a l a m o u n t o f ch a rg e d q = ¸ d y lo ca ted b etw een y a n d y + d y w ill lea d to a n electro sta tic p o ten tia l d q= r a t th e ¯ eld p o in t, w h ere r = (x 2 + y 2 )1 = 2 . S o th e to ta l p o ten tia l is g iv en by Z+ 1 Z+ 1 dy dy p p Á (x ) = ¸ : (1 ) = 2¸ x2 + y2 x 2 + y2 ¡1 0 C h a n g in g va ria b les fro m y to u = y = x , th e in teg ra l b eco m es Á (x ) = 2 ¸

Z+ 1 0

p

du : 1 + u2

(2 )

T h is resu lt is clea rly in d ep en d en t o f x a n d h en ce a co n sta n t w h ich is w h a t d im en sio n a l a n a ly sis to ld u s. M u ch w o rse, it is a n in ¯ n ite co n sta n t sin ce th e in teg ra l d iv erg es a t th e u p p er lim it. W h a t is g o in g o n in su ch a sim p le, cla ssic, tex tb o o k p ro b lem ? A s a ¯ rst a ttem p t in g ettin g a sen sib le resu lt, let u s cu to ® th e in teg ra l a t so m e len g th sca le y = ¤ . (Y o u m ay th in k o f th e in ¯ n ite lin e ch a rg e a s th e lim it o f a lin e ch a rg e o f len g th 2 ¤ w ith ¤ À x .) U sin g th e su b stitu tio n y = x sin h µ a n d ta k in g th e lim it ¤ À x , w e g et

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The problem has to

Á (x ) =

do with logarithms which allow a dimensionless function like ln (x/2) to occur in the electrostatic potential without the electric field depending on the arbitrary scale .

2¸

Z¤ 0

dy p = 2 ¸ sin h ¡1 x 2 + y2

µ ¶ ³x ´ ¤ ¼ ¡ 2 ¸ ln ; x 2¤ (3 )

w h ere w e h av e u sed ¤ À x in a rriv in g a t th e ¯ n a l eq u a lity. T h is p o ten tia l d o es d iv erg e w h en ¤ ! 1 . B u t n o te th a t th e p h y sica lly o b serva b le q u a n tity, th e electric ¯ eld E = ¡ r Á is in d ep en d en t o f th e cu t-o ® p a ra m eter ¤ a n d is co rrectly g iv en b y E x = 2 ¸ = x . B y in tro d u cin g a cu to ® , w e seem to h av e sav ed th e situ a tio n . It is n ow clea r w h a t is g o in g o n . A s th e title o f th is a rticle im p lies, th e p ro b lem h a s to d o w ith lo g a rith m s w h ich a llow a d im en sio n less fu n ctio n lik e ln (x = 2 ¤ ) to o ccu r in th e electro sta tic p o ten tia l w ith o u t th e electric ¯ eld d ep en d in g o n th e a rb itra ry sca le ¤ . T h is req u ires a d d itiv ity o n th e ¤ d ep en d en ce; th a t is w e n eed a fu n ctio n f (x = ¤ ) w h ich w ill red u ce to f (x ) + f (¤ ). C lea rly o n ly a lo g a rith m w ill d o . O n ce w e k n ow w h a t is h a p p en in g , it is ea sy to ¯ g u re o u t o th er w ay s o f g ettin g a sen sib le a n sw er. O n e ca n , fo r ex a m p le, o b ta in th is resu lt fro m a m o re stra ig h tfo rw a rd sca lin g a rg u m en t b y co n cen tra tin g o n th e p o ten tia l di® eren ce Á (x )¡ Á (a ), w h ere a is so m e a rb itra ry sca lin g d ista n ce w e in tro d u ce in to th e p ro b lem . F ro m d im en sio n a l a n a ly sis, it fo llow s th a t th e p o ten tia l d i® eren ce m u st h av e th e fo rm Á (x ) ¡ Á (a ) = ¸ F (x = a ), w h ere F is a d im en sio n less fu n ctio n . E va lu a tin g th is ex p ressio n fo r a = 1 , say, in so m e u n its w e g et ¸ F (x ) = Á (x ) ¡ Á (1 ). S u b stitu tin g b a ck , w e h av e th e rela tio n Á (x ) ¡ Á (a ) = Á (x = a ) ¡ Á (1 ). T h is fu n ctio n a l eq u a tio n h a s th e u n iq u e so lu tio n s Á (x ) = A ln x + Á (1 ). D im en sio n a l a n a ly sis a g a in tells y o u th a t A / ¸ . B u t, o f co u rse, sca lin g a rg u m en ts ca n n o t d eterm in e th e p ro p o rtio n a lity co n sta n t. H ow ev er, o n e ca n co m p u te th e p o ten tia l d i® eren ce b y th e ex p licit in teg ra l

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Á (x ) ¡ Á (a ) = 2 ¸

Z1 0

dy

Ã

1

1 p ¡ p x2 + y2 a2 + y2

!

:

(4 )

phenomena, in which naive scaling arguments break

It is ea sy to see th a t th is in teg ra l is ¯ n ite. Y o u ca n w o rk it o u t b y fa irly stra ig h tfo rw a rd p ro ced u res a n d o b ta in th e resu lt Á (x ) ¡ Á (a ) = ¡ 2 ¸ ln (x = a ):

It turns out that such

(5 )

T h e n u m erica l va lu e o f Á (x ) in th is ex p ressio n is in d ep en d en t o f th e len g th sca le a in tro d u ced in th e p ro b lem . In th a t sen se th e sca le o f Á is d eterm in ed o n ly b y ¸ w h ich , a s w e sa id b efo re, h a s th e co rrect d im en sio n s. B u t to en su re ¯ n ite va lu es fo r th e ex p ressio n s, w e n eed to in tro d u ce a n a rb itra ry len g th sca le a w h ich is th e k ey fea tu re I w a n t to em p h a size in th is d iscu ssio n . It tu rn s o u t th a t su ch p h en o m en a , in w h ich n a iv e sca lin g a rg u m en ts b rea k d ow n d u e to th e o ccu rren ce o f lo g a rith m ic fu n ctio n , is a v ery g en era l fea tu re in sev era l a rea s o f p h y sics esp ecia lly in th e stu d y o f ren o rm a liza tio n g ro u p in h ig h en erg y p h y sics. W h a t w e h av e h ere is a v ery elem en ta ry m a n ifesta tio n o f th is resu lt. In a ll th ese ca ses w e n eed to sm u g g le in to th e p ro b lem a len g th sca le to m a k e so m e u n o b serva b le q u a n tities (lik e th e p o ten tia l) ¯ n ite b u t a rra n g e m a tters su ch th a t o b serva b le q u a n tities rem a in in d ep en d en t o f th is sca le w h ich w e b rin g in .

down due to the occurrence of logarithmic function, is a very general feature in several areas of physics especially in the study of renormalization group in high energy physics.

If y o u th o u g h t th is w a s to o sim p le, h ere is a m o re so p h istica ted o ccu rren ce o f a lo g a rith m fo r essen tia lly th e sa m e rea so n . C o n sid er th e S ch rÄo d in g er eq u a tio n in tw o d im en sio n s fo r a n a ttra ctiv e D ira c d elta fu n ctio n p o ten tia l V (x ) = ¡ V 0 ± (x ) w ith V 0 > 0 . T h e v ecto r x is in tw o d im en sio n a l sp a ce a n d w e lo o k fo r a sta tio n a ry b o u n d sta te

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w av efu n ctio n à (x ) w h ich sa tis¯ es th e eq u a tio n µ ¶ ~2 2 ¡ r ¡ V 0 ± (x ) à (x ) = ¡ jE jà (x ); 2m

(6 )

w h ere ¡ jE j is th e n eg a tiv e b o u n d sta te en erg y. R esca lin g th e va ria b les b y in tro d u cin g ¸ = 2 m V 0 = ~2 a n d E = 2 m jE j= ~2 , th is eq u a tio n red u ces to ¡ 2 ¢ r + ¸ ± (x ) Ã (x ) = E Ã (x ): (7 )

W e co u ld h av e d o n e ev ery th in g u p to th is p o in t in a n y sp a tia l d im en sio n . In D d im en sio n , th e D ira c d elta fu n ctio n ± (x ) h a s th e d im en sio n L ¡ D . T h e k in etic en erg y o p era to r r 2 , o n th e o th er h a n d , a lw ay s h a s th e d im en sio n L ¡ 2 . T h is lea d s to a p ecu lia r b eh av io u r w h en D = 2 . W e ¯ n d th a t, in th is ca se, ¸ is d im en sio n less w h ile E h a s th e d im en sio n o f L ¡ 2 . S in ce th e sca led b in d in g en erg y E h a s to b e d eterm in ed en tirely in term s o f th e p a ra m eter ¸ , w e h av e a p ro b lem in o u r h a n d s. T h ere is n o w ay w e ca n d eterm in e th e fo rm o f E w ith o u t a d im en sio n a l co n sta n t { w h ich w e d o n o t h av e. T o see th e m a n ifesta tio n o f th is p ro b lem m o re clea rly, let u s so lv e (7 ). T h is is fa irly ea sy to d o b y F o u rier tra n sfo rm in g b o th sid es a n d in tro d u cin g th e m o m en tu m sp a ce w av efu n ctio n Á (k ) b y Z Á (k ) = d 2 x à (x ) ex p (¡ ik ¢ x ): (8 ) T h e left-h a n d sid e o f lea d s to th e term [¡ k 2 Á (k )+ ¸ à (0 )], w h ile th e rig h t-h a n d sid e g iv es E Á (k ). E q u a tin g th e tw o w e g et ¸ à (0 ) Á (k ) = 2 : (9 ) k + E W e n ow in teg ra te th is eq u a tio n ov er a ll k . T h e left-h a n d sid e w ill th en g iv e (2 ¼ )2 à (0 ) w h ich ca n b e ca n celled o u t o n b o th sid es b y a ssu m in g à (0) 6= 0 . (T h is is, o f co u rse,

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n eed ed fo r Á (k ) in (9 ) to b e n o n zero a n d h en ce is n o t a n a d d itio n a l a ssu m p tio n .) W e th en g et th e resu lt 1 1 = ¸ 4¼ 2

Z

d2k 1 = 2 k + E 4¼ 2

Z

d2s : s2 + 1

The Dirac delta function, in spite of the nomenclature,

(1 0 )

is strictly not a function but what

T h e seco n d eq u a lity is o bp ta in ed b y ch a n g in g th e in teg ra tio n va ria b le to s = k = E . T h is eq u a tio n is su p p o sed to d eterm in e th e b in d in g en erg y E in term s o f th e p a ra m eter in th e p ro b lem ¸ b u t th e la st ex p ressio n sh ow s th a t th e rig h t h a n d sid e is in d ep en d en t o f E ! T h is is sim ila r to th e situ a tio n in th e electro sta tic p ro b lem in w h ich w e g o t th e in teg ra l w h ich w a s in d ep en d en t o f x . In fa ct, ju st a s in th e electro sta tic ca se, th e in teg ra l o n th e rig h t h a n d sid e d iv erg es, co n ¯ rm in g o u r su sp icio n . O f co u rse, w e a lrea d y k n ow th a t d eterm in in g E in term s o f ¸ is im p o ssib le d u e to d im en sio n a l m ism a tch .

mathematicians call a distribution.

O n e ca n , a t th is sta g e, ta k e th e p o in t o f v iew th a t th e p ro b lem is sim p ly ill-d e¯ n ed a n d o n e w o u ld b e q u ite co rrect. T h e D ira c d elta fu n ctio n , in sp ite o f th e n o m en cla tu re, is strictly n o t a fu n ctio n b u t w h a t m a th em a ticia n s ca ll a d istrib u tio n . It is d e¯ n ed a s a lim it o f a seq u en ce o f fu n ctio n s. F o r ex a m p le, su p p o se w e co n sid er a seq u en ce o f p o ten tia ls · ¸ V0 jx j2 V (x ) = ¡ ; ex p ¡ 2¼ ¾ 2 2¾ 2

(1 1 )

w h ere x is a 2 -D v ecto r a n d ¾ is a p a ra m eter w ith th e d im en sio n o f len g th . In th is ca se, w e w ill a g a in g et (7 ) b u t w ith th e D ira c d elta fu n ctio n rep la ced b y th e G a u ssia n in (1 1 ). B u t n ow w e h av e a p a ra m eter ¾ w ith th e d im en sio n o f len g th a n d o n e ca n im a g in e th e b in d in g en erg y b ein g co n stru cted o u t o f th is. W h en w e ta k e th e lim it ¾ ! 0 , th e p o ten tia l in (1 1 ) red u ces to o n e p ro p o rtio n a l to th e D ira c d elta fu n ctio n . (T h is is w h a t w e m ea n t b y say in g th e d elta fu n ctio n is d e¯ n ed a s a lim itin g ca se o f seq u en ce o f fu n ctio n s. H ere th e fu n ctio n s a re G a u ssia n s

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The essential idea is to accept that the theory requires an extra scale with proper dimensions for its interpretation and treat the coupling constant a function of the scale at which we probe the system.

in (1 1 ) p a ra m etrized b y ¾ . W h en w e ta k e th e lim it o f ¾ ! 0 th e fu n ctio n red u ces to d elta fu n ctio n .) T h e tro u b le is th a t, w h en w e let ¾ g o to zero , w e lo se th e len g th sca le in th e p ro b lem a n d w e d o n o t k n ow h ow to ¯ x th e b in d in g en erg y. O f co u rse, n o o n e a ssu red y o u th a t if y o u so lv e a d i® eren tia l eq u a tio n w ith a n in p u t fu n ctio n V (x ;¾ ) w h ich d ep en d s o n a p a ra m eter ¾ a n d ta k e a (so m ew h a t d u b io u s) lim it o f ¾ ! 0 , th en th e so lu tio n s w ill a lso h av e a sen sib le lim it. S o o n e ca n say th a t th e p ro b lem is ill-d e¯ n ed . R a th er th a n leav in g it a t th a t, w e w a n t to a ttem p t h ere so m eth in g sim ila r to w h a t w e d id in th e electro sta tic ca se. L et u s eva lu a te th e in teg ra l w ith a cu t-o ® a t so m e va lu e k m a x = ¤ w ith ¤ 2 À E . T h en w e g et µ ¶ E 1 1 ; = ¡ ln (1 2 ) ¸ 4¼ ¤2 w h ich ca n b e in v erted to g iv e th e b in d in g en erg y to b e: E = ¤ 2 ex p (¡ 4 ¼ = ¸ );

(1 3 )

w h ere th e sca le is ¯ x ed b y th e cu t-o ® p a ra m eter. O f co u rse th is is w h a t w e w o u ld h av e g o t if w e a ctu a lly u sed a p o ten tia l w ith a len g th sca le. T h ere is a w a y o f in terp retin g th is resu lt ta k in g a cu e fro m w h a t is d o n e in q u a n tu m ¯ eld th eo ry. T h e essen tia l id ea is to a ccep t u p fro n t th a t th e th eo ry req u ires a n ex tra sca le w ith p ro p er d im en sio n s fo r its in terp reta tio n a n d trea t th e co u p lin g co n sta n t a s a fu n ctio n o f th e sca le a t w h ich w e p ro b e th e sy stem . H av in g d o n e th a t w e a rra n g e m a tters so th a t th e o b serv ed resu lts a re a ctu a lly in d ep en d en t o f th e sca le w e h av e in tro d u ced . In th is ca se, w e w ill d e¯ n e a p h y sica l co u p lin g co n sta n t b y µ ¶ E 1 1 ¡1 ¡1 2 2 ¸ p h y (¹ ) = ¸ ¡ ; (1 4 ) ln (¤ = ¹ ) = ¡ ln ¹2 4¼ 4¼

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w h ere ¹ is a n a rb itra ry b u t ¯ n ite sca le. O b v io u sly ¸ p h y (¹ ) is in d ep en d en t o f th e cu t-o ® p a ra m eter ¤ . T h e b in d in g en erg y is n ow g iv en b y E = ¹ 2 ex p (¡ 4 ¼ = ¸ p h y (¹ ))

(1 5 )

w h ich , in sp ite o f a p p ea ra n ce, is in d ep en d en t o f th e sca le ¹ . T h is is sim ila r to o u r eq u a tio n (5 ) in th e electro sta tic p ro b lem , in w h ich w e in tro d u ced a sca le a b u t Á (x ) w a s in d ep en d en t o f a .

The breaking down of naive scaling arguments and the appearance of logarithms are rather ubiquitous in such a case.

In q u a n tu m ¯ eld th eo ry a resu lt lik e th is w ill b e in terp reted a s fo llow s: S u p p o se o n e p erfo rm s a n ex p erim en t to m ea su re so m e o b serva b le q u a n tity (lik e th e b in d in g en erg y ) o f th e sy stem a s w ell a s so m e o f th e p a ra m eters d escrib in g th e sy stem (like th e co u p lin g co n sta n t). If th e ex p erim en t is p erfo rm ed a t a sca le co rresp o n d in g to ¹ (w h ich , fo r ex a m p le, co u ld b e th e en erg y o f th e p a rticles in a sca tterin g cro ss-sectio n m ea su rem en t, say ), th en o n e w ill ¯ n d th a t th e co u p lin g co n sta n t th a t is m ea su red d ep en d s o n ¹ . B u t w h en o n e va ries ¹ in a n ex p ressio n lik e (1 5 ), th e va ria tio n o f ¸ p h y w ill b e su ch th a t o n e g ets th e sa m e va lu e fo r E . W h en y o u th in k a b o u t it, y o u w ill ¯ n d th a t it m a k es a lo t o f sen se. A fter a ll th e p a ram eters w e u se to d escrib e o u r p h y sica l sy stem (lik e ¸ p h y ) a s w ell a s so m e o f th e resu lts w e o b ta in (lik e th e b in d in g en erg y E o r a sca tterin g cro ss-sectio n ) n eed to b e d eterm in ed b y su ita b le ex p erim en ts. In th e q u a n tu m m ech a n ica l p ro b lem s a lso o n e ca n th in k o f sca tterin g of a p a rticle w ith m o m en tu m k (rep resen ted b y a n in cid en t p la n e w av e, say ) b y a p o ten tia l. T h e resu ltin g scatterin g cro ss-sectio n w ill co n ta in in fo rm a tio n a b o u t th e p o ten tia l, esp ecia lly th e co u p lin g co n sta n t ¸ . If th e scatterin g ex p erim en t in tro d u ces a (m o m en tu m o r len g th ) sca le ¹ , th en o n e ca n in d eed im a g in e th e m ea su red co u p lin g co n sta n t to b e d ep en d en t o n th a t sca le ¹ . B u t w e w o u ld ex p ect p h y sica l p red ictio n s o f th e th eo ry (lik e E ) to b e in d ep en d en t

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o f ¹ . T h is is p recisely w h a t h a p p en s in q u a n tu m ¯ eld th eo ry a n d th e toy m o d el a b ov e is a sim p le illu stra tio n . W e see fro m (7 ) th a t, in D = 1 , th e co u p lin g co n sta n t ¸ h a s th e d im en sio n s o f L ¡1 so th ere is n o d i± cu lty in o b ta in in g E / ¸ 2 . T h e o n e-d im en sio n a l in teg ra l co rresp o n d in g to (1 0 ) is co n v erg en t a n d y o u ca n ea sily w o rk th is o u t to ¯ x th e p ro p o rtio n a lity co n sta n t to b e 1 / 4 . T h e lo g a rith m ic d iv erg en ce o ccu rs in D = 2 , w h ich is k n ow n a s th e critica l d im en sio n fo r th is p ro b lem . T h e b rea k in g d ow n o f n a iv e sca lin g a rg u m en ts a n d th e a p p ea ra n ce o f lo g a rith m s a re ra th er u b iq u ito u s in su ch a ca se. (T h ere a re o th er fa scin a tin g issu es in D ¸ 3 a n d in sca tterin g b u t th a t is a n o th er sto ry.)

Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]

T h e ex a m p les d iscu ssed h ere a re a ll ex p lo red ex ten siv ely in th e litera tu re a n d a g o o d sta rtin g p o in t w ill b e th e referen ces [1 -5 ]. Suggested Reading [1] L R Mead and J Godines, Am. J. Phys., Vol.59, No.10, pp.935–937, 1991. [2] P Gosdzinsky and R Tarrach, Am. J. Phys., Vol.59, No.1, pp.70–74 1991. [3] B R Holstein, Am. J. Phys., Vol.61, No.2, pp.142–147, 1993. [4] A Cabo, J L Lucio and H Mercado, Am. J. Phys., Vol.66, No.3, pp.240– 246, 1998. [5] M Hans, Am. J. Phys., Vol.51, No.8, pp.694–698, 1983.

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