Physics 506 Winter 2006 Homework Assignment #12 Solutions. Textbook problems: Ch. 14: 14.2, 14.4, 14.6, 14.12

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1 Physis 56 Wintr 6 Homwork Assignmnt # Solutions Ttbook problms: Ch. 4: 4., 4.4, 4.6, A partil of harg is moving in narly uniform nonrlativisti motion. For tims nar t = t, its vtorial position an b pandd in a Taylor sris with fid vtor offiints multiplying powrs of (t t ). a) Show that, in an inrtial fram whr th partil is instantanously at rst at th origin but has a small alration a, th Liénard-Wihrt ltri fild, orrt to ordr / inlusiv, at that instant is E = E v + E a, whr th vloity and alration filds ar E v = ˆr r + [ a 3ˆr(ˆr a )]; r Ea = [ a ˆr(ˆr a )] r and that th total ltri fild to this ordr is E = ˆr r [ a + ˆr(ˆr a )] r Th unit vtor ˆr points from th origin to th obsrvation point and r is th magnitud of th distan. Commnt on th r dpndns of th vloity and alration filds. Whr is th pansion likly to b valid? Epanding th position around a tim t givs r(t ) = r + v(t t ) + a(t t ) + Howvr, w work in th instantanous rst fram with th partil at th origin. Hn it is suffiint to onsidr r(t ) = a(t t ) +, β(t ) = a(t t ) +, β(t ) = a + To prod, w would lik to dvlop a rlation btwn obsrvr tim t and rtardd tim t. Th at prssion is of ours t = t + r(t ) /. Howvr, sin w wish to pand at tim t t, it is suffiint to writ t = t + / + whr =. Th omittd trms turn out to b of highr ordr in /. W now writ down th ltri fild at obsrvr tim t = t. This orrsponds to a rtardd tim t = t /. As a rsult, th various prssions showing up in th vloity and alration filds ar givn (up to ordr / ) by r = a, β = a, β = a

2 as wll as R = r = a R = ( ˆ a ), ˆn R R = ˆ [ a ˆ(ˆ a )] () W also not that /γ = β = + O(/ 4 ) = + to th ordr of intrst. This yilds th filds ˆn β E v (, t ) = γ R ( β ˆn) = 3 ˆ [ a ˆ(ˆ a )] + a ( ˆ a ) ( + ˆ a ) 3 = ˆ + ([ a + ˆ(ˆ a )] ( + (ˆ a )) = ˆ + [ a 3ˆ(ˆ a )] whih agrs with th dsird rsult (although w hav usd and ˆ instad of r and ˆr). Th rsult for th alration fild is vn mor straightforward, as th lading trm is alrady of ordr / E a (, t ) = Adding () and (3) givs ˆn [(ˆn β) β] R( β = ˆ (ˆ a ) ˆn) 3 E = ˆ [ a + ˆ(ˆ a )] = ˆ (ˆ a ) = [ a ˆ(ˆ a )] Not that th vloity fild ontains th stati Coulomb trm ˆ/ along with an alration trm, whih is prhaps unusual for a vloity fild. Th lattr only falls of as / for larg, whih is also surprising, as th vloity fild ordinarily is thought of as a /R fild. Th alration fild is as ptd, howvr, as it dpnds on alration and hibits th propr /R bhavior. Th rsolution to this apparnt disrpany is th fat that our pansion is only valid for small valus of, namly /a, whr th rtardd tim approimation is valid (orrsponding to th / trm in () bing small ompard to th lading trm). Roughly this is similar to bing in th nar on (and not th radiation on). b) What is th rsult for th instantanous magnti indution B to th sam ordr? Commnt. Th magnti indution is givn by B = ˆn E = ( [ a ˆ(ˆ a )] ) = ( a ˆ + ˆ a ) = ( ˆ ) [ a + ˆ(ˆ a )] () (3)

3 In othr words, th instantanous B vanishs (to this lvl of approimation). This should not b surprising, baus th partil is instantanously at rst (and a stati partil dos not gnrat a magnti fild). ) Show that th / trm in th ltri fild has ro divrgn and that th url of th ltri fild is E = (ˆr a )/ r. From Faraday s law, find th magnti indution B at tims nar t =. Compar with th familiar lmntary prssion. W omput th divrgn as follows ( ) ( ) ( ) a + ˆ(ˆ a ) = a + ( a ) ( ) ( ) 3 = a + 3 ( a ) + 3 ( ( a ) ) = ˆ a 3 ˆ a + 4 ˆ a = This dmonstrats that th / trm has ro divrgn. For th url, w obtain ( ) E = 3 ( ) a ( a ) + 3 = ( ( ) a + ) ( a ) ˆ = ( ˆ a + a ˆ ) = Faraday s law stats E + (/) B/ t =. Hn Intgrating this for tims nar t givs B = B t = E = ˆ a ˆ [ a(t t )] = ˆ v(t) ˆ a = v(t) ˆ This rprodus th lmntary Biot-Savart law for th magnti fild. 4.4 Using th Liénard-Wihrt filds, disuss th tim-avragd powr radiatd pr unit solid angl in nonrlativisi motion of a partil with harg, moving a) along th ais with instantanous position (t) = a os ω t. In th non-rlativisiti limit, th radiatd powr is givn by dp (t) dω = 4π ˆn β (4)

4 In th as of harmoni motion along th ais, w tak r = ẑa os ω t, β = ẑ aω sin ω t, β = ẑ aω os ω t By symmtry, w assum th obsrvr is in th - plan tiltd with angl θ from th vrtial. In othr words, w tak ˆn = ˆ sin θ + ẑ os θ This provids nough information to simply substitut into th powr prssion (4) ˆn β = ŷ aω sin θ os ω t dp (t) dω = a ω 4 4π 3 sin θ os ω t Taking a tim avrag (os ω t ) givs dp dω = a ω 4 8π 3 sin θ This is a familiar dipol powr distribution, whih looks lik Intgrating ovr angls givs th total powr P = a ω b) in a irl of radius R in th -y plan with onstant angular frquny ω. Skth th angular distribution of th radiation and dtrmin th total powr radiatd in ah as. Hr w tak instad r = R(ˆ os ω t + ŷ sin ω t) β = Rω ( ˆ sin ω t + ŷ os ω t) β = Rω (ˆ os ω t + ŷ sin ω t)

5 Thn ˆn Rω β = [ŷ os θ os ω t + (ẑ sin θ ˆ os θ) sin ω t] whih givs dp (t) dω = R ω 4 4π 3 (os θ os ω t + sin ω t) Taking a tim avrag givs dp dω = R ω 4 8π 3 ( + os θ) This distribution looks lik Th total powr is givn by intgration ovr angls. Th rsult is P = R ω a) Gnrali th irumstans of th ollision of Problm 4.5 to nonro angular momntum (impat paramtr) and show that th total nrgy radiatd is givn by W = 4 ( m ) / ( ) ) / dv (E 3m 3 V (r) L r min dr mr dr whr r min is th losst distan of approah (root of E V L /mr ), L = mbv, whr b is th impat paramtr, and v is th inidnt spd (E = mv /). In th non-rlativisti limit, w may us Lamour s formula writtn in trms of p P (t) = 3m 3 d p dt = ( ) dv (r) 3m 3 (5) dr whr w notd that th ntral potntial givs a for d p/dt = F = ˆrdV/dr. Th radiatd nrgy is givn by intgrating powr ovr tim W = P (t) dt

6 Howvr, this an b onvrtd to an intgral ovr th trajtory. By symmtry, w doubl th valu of th intgral from losd approah to infinity W = r min P dr (6) dr/dt Th radial vloity dr/dt an b obtaind from nrgy onsrvation E = mṙ + ) / L mr + V (r) dr (E dt = V (r) L m mr Substituting P (t) from (5) as wll as dr/dt into (6) thn yilds W = 4 m ( ) ) / dv (E 3m 3 V (r) L r min dr mr dr (7) b) Spiali to a rpulsiv Coulomb potntial V (r) = Z /r. Show that W an b writtn in trms of impat paramtr as ( ) ] E = mv5 [ t 4 Z 3 + t 5 + t tan t 3 whr t = bmv /Z is th ratio of twi th impat paramtr to th distan of losst approah in a had-on ollision. Substituting into (7) givs V (r) = Z, L = mbv, E = r mv W = 44 Z 6 3m 3 v = 4mv5 3Z 3 t 3 = 4mv5 3Z 3 t 3 = 4mv5 3Z 3 t 3 r 4 r min r min ma + ( b r ) / ( Z mv r b r dr ) ( b ) / tr b b dr r r (/t) d ( )( + ) d (8) whr w usd t = bmv /Z, and th variabl substitution = b/r. In th last lin + and ar th roots of th quadrati quation ± = t ± t +

7 Th intgral an b prformd by Eulr substitution. W us th indfinit intgral ( )( + ) d = ( )( + ) ( + 3( + + ) ) 4 Putting in limits givs ( 3(+ + ) 4 + ) tan + ( )( + ) d = ( + + ) ( + 4 3(+ + ) ) 4 + tan + = 3 ( ) ( 3 t + t + tan ) t + t + Th artan trm an b simplifid by doubl angl rlations to giv + ( )( + ) d = 3 t + ( ) 3 t + tan t Insrting this into (8) finally givs W = mv5 Z 3 ( t 4 + t 5 ( + t 3 ) ) tan t 4. As in Problm 4.4a), a harg movs in simpl harmoni motion along th ais, (t ) = a os(ω t ). a) Show that th instantanous powr radiatd pr unit solid angl is whr β = aω /. dp (t ) dω = β 4 4πa sin θ os (ω t ) ( + β os θ sin ω t ) 5 For on-dimnsional motion, th rlativisti radiatd powr prssion simplifis dp (t ) dω = ˆn [(ˆn β) β ] 4π ( β = ˆn (ˆn β ) ˆn) 5 4π ( β = ˆn β ˆn) 5 4π ( β ˆn) (9) 5 W us th sam stup as Problm 4.a), namly r = ẑa os ω t, β = ẑ aω sin ω t, β = ẑ aω os ω t

8 Th obsrvr is loatd at a point whih givs ris to or = (ˆ sin θ + ẑ os θ) R = r = ˆ sin θ + ẑ( os θ a os ω t ) R = ( a ) / os θ os ω t + a os ω t R, ˆn = R This rathr ompliatd prssion atually simplifis in th radiation on ( ), whih is th only rgion w ar intrstd in. In this as, R = and ˆn = / = ˆ sin θ + ẑ os θ. Noting that w simply valuat (9) to obtain β ˆn = + aω os θ sin ω t dp (t ) dω = a ω 4 sin θ os ω t 4π 3 ( + aω os θ sin ω t ) 5 Making th substitution β = aω / thn rsults in dp (t ) dω = β 4 4πa sin θ os ω t ( + β os θ sin ω t ) 5 () b) By prforming a tim avraging, show that th avrag powr pr unit solid angl is dp dω = β 4 [ 4 + β os ] θ 3πa sin θ ( β os θ) 7/ To tim avrag (), w nd to prform th intgral = I(a) = π os α ( + a sin α) 5 dα This may b prformd by ompl variabls thniqus. Dfining = iα onvrts this to a ontour intgral I(a) = 8 ( + ) a 5 ( + i/a ) 5 d = 8 ( + ) a 5 ( ) 5 ( + ) 5 d whr + and ar th roots = ± = i a ± i a

9 It is asy to s that only + lis insid th unit irl, providd < a <. (Sin I( a) = I(a), w an tnd th rsult to a <.) As a rsult, th valu of I(a) oms from th rsidu at + I(a) = 6πi d 4 a 5 4! d 4 Working out th drivativs givs th rsult I(a) = π 4 ( ) ( + ) ( ) a ( a ) 7/ Using a = β os θ for tim avraging (), w find = + dp dω = β β os θ 3πa ( β os θ) 7/ sin θ ) Mak rough skths of th angular distribution for nonrlativisti and rlativisti motion. Th nonrlativisti limit yilds ordinary dipol radiation. Th angular distribution for various valus of β ar β = β = β =.7 β = β =.98 β =.99 Th rlativisti baming fft (along th ais) is larly pronound at larg valus of β.

Lecture 14 (Oct. 30, 2017)

Lecture 14 (Oct. 30, 2017) Ltur 14 8.31 Quantum Thory I, Fall 017 69 Ltur 14 (Ot. 30, 017) 14.1 Magnti Monopols Last tim, w onsidrd a magnti fild with a magnti monopol onfiguration, and bgan to approah dsribing th quantum mhanis