DO NOT WRITE YOUR NAME OR STUDENT NUMBER ON ANY SHEET!

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Edith Carroll
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1 Electodynamics Subject Exam Pep Quiz Fiday Apil DO NOT WRITE YOUR NAME OR STUDENT NUMBER ON ANY SHEET! E&MStudyGuide Ch 1 Special Relativity Pime 1.1 Gamma Factos and Such tlab γt0 Llab L0 γ 1.2 Loentz Tansfomations α L α β β L α β γ γv 0 0 γv γ fo ˆx boost γ 2 (1 v 2 )1 sinhη γv coshη γη apidity 1. invaiants and the Metic Tenso g αβ Dot poduct: A α gαβb β A α Bα invaiant gαβ g αβ µ x µ Continuity Eqn: J 0 Maxwell s Eqns: αf αβ J β α F αβ Fou-Velocities and Momenta u (γ γ d x dt u α uα 1 p mu p α pα m 2 Ch 2 Dynamics of a Relativistic Point 2.1 Lagangian fo a Fee Relativistic Paticle S m dt ( dt dt )2 ( d dt )2 α α + δω αβ (t)β δs dt L d α dt (δωαβ β ) dt[ d dt (πα β)+παṙβ]δω αβ 0 d dt (α p β p α β )0 2.2 Inteaction of a Chaged Paticle with an Extenal EM Field S dτ( m eu A) m dt ( dt dt )2 ( d dt )2 e dt(a0 v A) L m 1 v 2 e((a0 v A) d dt (γmvi) e ia0 e tai + e v ( A) E A0 t A B A d dt (γm v)e E + e v B 2. Motion in a Constant Magentic Field A xbŷ B Bẑ S dt[m 1 v 2 exbvy] d dt (γm v)e(vy ˆx vxŷ)b vx eb γm vy ωvy vy eb γm vx ωvx vx ωrsin(ωt) x Rcos(ωt) vy ωrcos(ωt) y Rsin(ωt) Gauge Tansfomations A µ A µ + µ Λ(t x y z) E B do not change 2.5 The EM Field Tenso F αβ α A β β A α Fαβ Ex 0 Bz By 0 Ex Ey Ez Ey Bz 0 Bx Ez By Bx 0 F αβ 0 Ex Ey Ez Ex 0 Bz By Ey Bz 0 Bx Ez By Bx 0 F α β 0 Ex Ey Ez Ex 0 Bz By Ey Bz 0 Bx Ez By Bx 0 d dτ uα ef αβ uβ Ch Dynamics of EM Fields.1 Lagangian (Density) fo Fee Fields: Deiving Maxwell s Equations Sf d 4 L(a µ νa µ ) µ L µa ν L A ν L 1 16π F µν Fµν 1 16π F µν Fνµ Sm d 4 J A µf µν 4πJ ν J α 1 Ω a Ω qa uα a Ωis4Dvolume γa E 4πJ 0 ( B) t E 4π J Dual EM tenso αβ 1 F 2 ϵµναβ Fαβ 0 Bx By Bz Bx 0 Ez Ey By Ez 0 Ex Bz Ey Ex 0 µ F µν 0 B 0 t B + E 0.2 Pseudo-Vectos and Pseudo-Scales F αβ Fαβ 4 E B pseudoscala F αβ Fαβ 2( E 2 B 2 )(egula)scala The Stess-Enegy Tenso of the EM Field T αβ pi α β φ g αγ L φ A µ π α L ( αφ) T αγ 1 4π F αβ F γ β + F µν Fµν T π (E2 + B 2 ) T 0i 1 4π ϵ ijk(ejbk) T ij 1 8π (δij(e2 + B 2 ) 2EiEj 2BiBj) U (f) 1 2 d A0( )J0( ).4 Hype-Sufaces and Consevation of E P L stess-enegy tenso is symmetic

2 Electostatics ~E 4 ~E 2 4 PE 1 d (~ ) (~ ) (~ ) Laplace Equation: 2 0 Laplace @ ˆ + 1 Solution @ ˆ + 1 sin @ sin 2 Solution Spheical Coodinates: ( ( ) X lm ) X l h A lm l + B lm (l+1)i Y m l ( ) h A l l + B l (l+1)i P l (cos ) 1

3 Othogonality elations: 1 1 (Yl m ) Yl m0 0 d ll 0 mm 0 2 P l (x)p l 0(x) dx ll 0 2l +1 Spheical hamonics of ode apple 2: Y Y e 2i sin 2 Y e i sin Y e i sin cos Y cos Y ( cos2 1) Y ei sin Y ei sin cos Y e2i sin 2 Legende polynomials of ode apple : P 0 1 P 1 x P (x2 1) P 1 2 (5x x) Multipole Expansions V mon () 1 Q 4 o V dip () 1 p ˆ 4 o 2 (~ ) X lm 4 (2l + 1) q Y lm ( ) lm l+1 Multipole Moments 1 q 00 4 q q (Q 11 2iQ 12 Q 22 ) q 11 8 (p x ip y ) q (Q 1 iq 2 ) 5 q 10 4 p z q Q 2

4 Monopole Moment Dipole moments q p i d (~ ) (1) d (~ ) i (2) Quadupole moments Q ij d (~ )( i j 2 ij) () Magnetostatics Vecto Potential Biot-Savat Law ~A(~ ) ~B(~ ) d 0 J(~ ~ 0 1 ) ~ ~ 0 (4) d 0 ~ J(~ 0 ) (~ ~ 0 ) ~ ~ 0 (5) Net foce on a cuent density J ~ due to a magnetic field ~F d J ~ B ~ (6) Magnetic Moment ~m 1 2 d ~ ~ J (7) Magnetic field due to magnetic moment Toque due to magnetic field Potential enegy of magnetic moment in a magnetic field Fo intinsic spin ~ S the magnetic moment is ~B ~m + ~ (~m ~ ) (8) 5 ~ ~m ~ B (9) U ~m ~B (10) Useful g factos: ~µ g e~ S 2m g ~ S h ( e h 2m ) (11) electon muon poton neuton

5 ' " LONG ANSWER SECTION 1. Conside a thin spheical shell of adius R with chage density σ σ 0 cos θ whee θ is the pola angle measued elative to the z axis. (a) (5 pts) State the conditions elating the electic potential Φ on opposite sides of the shell R ϵ and R 1 ϵ. (b) (10 pts) Solve fo the potential fo both <Rand >R. (c) (5 pts) What is the electic dipole moment d? III. + b) of e- { Auso t.ie#ooaso. Base / - R < > o 2B/ 4IG Bli AR g- { A B + A 4T Go / 4nF Gotts 41T. it < R > R a) d+ds - 0 dzz#fr' dwso 2 IT R6oztn a co so 4T6oR. Rae

6 SECRET STUDENT NUMBER: STUDNUMBER Exta wokspace fo #1

7 ' 2. A squae loop of dimensions L L is centeed at the oigin with its plane pointing along the z axis. Two chages each +Q move at a fixed speed v<<c counte-clockwise aound the loop as viewed looking down on the x y plane fom the +z diection. The chages always ae positioned on opposite sides of the loop. (a) (5 pts) What is the time-aveaged magnetic dipole moment? (b) (10 pts) What is the magnetic field at a position (x 0y 0z) as a function of time. L a) I 2 Q It vt4 + 44v Mz Qz L E. A Qvlz b) At time t I < z t R two Ian FEE < Yz fiditq ^ B - chaise one all in Repeat vql fun ehmsect Btf( ie+at.vlq[c± zitzgsn tpmtm " B- I EBit- ym ' + malting ^ by # 2 pm < E othe chap then epeat afte 0tf

8 SECRET STUDENT NUMBER: STUDNUMBER Exta wokspace fo #2

9 . (10 pts) Conside a finite volume with some constant cuent density J( ). Using cuent consevation J 0 pove that d J( )0. F) 0 - fai fai - fai J ; 2kJn ) fni fot J

10 Exta wok space fo #

11 4. YES o NO (4 pts) (a) A configuation of chages has no electic monopole o dipole moments but does have an electic quadupole moment. Will the quadupole field as seen by an obseve fa away be changed by moving the entie distibution by a finite displacement a? NO (b) A configuation of chages has no electic monopole but does have a non-zeo electic dipole moment and a non-zeo electic quadupole moment. Will the dipole field as seen by an obseve fa away be changed by moving the entie distibution by a finite displacement a? (c) A configuation of chages has no electic monopole but does have a non-zeo electic dipole moment and a non-zeo electic quadupole moment. Will the quadupole field as seen by an obseve fa away be changed by moving the entie distibution by a finite displacement a? :ES a) R b) c) 5. (4 pts) Conside the thee chage configuations shown above: (a) two point chages +Q sepaated by R (b) two sphees of adius < R/2 each with chage +Q unifomly spead thoughout the volume with the centes sepaated by R (c) two spheical shells of adius < R/2 each with chage +Q unifomly spead thoughout the suface with the centes sepaated by R. The wok equied to move the sphees (o points) fom infinity to the sepaation R is labeled W a W b and W c fo each configuation. Label each statement as tue o false. (a) W a >W b (b) W a >W c (c) W b >W c false false false

Electrodynamics Subject Exam Prep Quiz, Tuesday, May 2, 2017

Electrodynamics Subject Exam Prep Quiz, Tuesday, May 2, 2017 Electodynamics ubject Exam Pep Quiz, Tuesday, May, 017 a ( b c ) b( a c ) c ( a b), a ( b c ) b ( c a ) c ( a b), ( a b) ( c d) ( a c )( b d) ( a d)( b c ), ( ψ) 0, ( a ) 0, ( a ) ( a ) a, (ψ a) a ψ +