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

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Harvey Brendan Warren
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1 tatistical Mechanics ubject Exam, Wednesday, May 3, 2017 DO NOT WRITE YOUR NAME OR TUDENT NUMBER ON ANY HEET! 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) 2 a, (ψ a) = a ψ + ψ a, (ψ a) = ψ a + ψ a, ( a b) = ( a ) b + ( b ) a + a ( b) + b ( a), ( a b) = b ( a) a ( b), ( a b) = a( b) b( a) + ( b ) a ( a ) b, r = 3, r = 0, ˆr = 2/r, ˆr = 0, ( a )ˆr = 1 r [ a ˆr( a ˆr)] = a r. V d 3 r A = V d 3 r ψ = V d 3 r A = V d 3 r (ϕ 2 ψ + ϕ ψ) = d 3 r (ϕ 2 ψ ψ 2 ϕ) = V ( A) d = d ψ = C 2 = 2 r + 2 r r d A, ψ d, d A, ϕ d ψ, (ϕ ψ ψ ϕ) d, d l A, d l ψ. 2 = ρ ρ ρ m2 r 2, ( ) 1 2 = δ( r). r l(l + 1) r 2,

2 L α β = γv γ 0 0 γ γv Fαβ = 0 Ex Ey Ez Ex 0 Bz By 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. m d dτ uα = ef αβ uβ, d p dt = e E + e v B, ωc = eb γm, E = J 0, ( B) t E = J, B = 0, t B + E = 0, αf αβ = J β, α F αβ = 0, e 2 = ħc , T αβ = π α β ϕ g αβ L, π α L ( αϕ), T 00 = 1 8π (E2 + B 2 ), T 0i = 1 ϵ ijkejbk, T ij = T i j = 1 8π (δ ij(e 2 + B 2 ) 2EiEj 2BiBj), E = A 0 t A = Φ t A, B = A. Pl(cos θ) = Y0,0 = Y1,±1 = Y2,±1 = 2l + 1 Y lm=0(θ), 1, Y1,0 = 3 cos θ, 3 8π sin θei±ϕ, Y2,0 = 15 8π sin θ cos θe±iϕ, Y2,±2 = 5 16π (3 cos2 θ 1), 15 32π sin2 θe ±2iϕ, Yl m(θ, ϕ) = ( 1) m Y lm(θ, ϕ), δll δ mm = dω Yl,m(θ, ϕ)yl,m (θ, ϕ), P0(x) = 1, P1(x) = x, P2(x) = 1 2 (3x2 1), P3(x) = 1 2 (5x3 3x), Pl(x = 1) = 1, (3D) Φ = lm 1 (2D) Φ = A0 ln(ρ) + m Φ = A0J0 = m Φ = q r E 1 = r Φ(r, θ, ϕ) = lm q22 = q21 = q20 = Qij dx Pl(x)Pl (x) = 2 1 2l + 1 δ ll, ( A lmr l + Blmr l 1) Ylm(θ, ϕ)e imϕ, p r + r 3 + m 15 32π 3 p + 3 p r 15 8π 5 16π e imϕ ( Amρ m + Bmρ m), e imϕ (AmJm(kρ) + BmNm(kρ)) e ±kz, 5r 3 q 2m(r)Y2m(θ, ϕ), r 5 r +, (2l + 1)r l+1 q lm(r)ylm(θ, ϕ), d 3 r ρ( r)(x iy) 2 = d 3 r ρ( r)(x iy)z = 5 d 3 r ρ( r)(3z 2 r 2 ) = 16π Q 33, d 3 r (3rirj r 2 δij)ρ( r), π (Q 11 2iQ12 Q22), 15 72π (Q 13 iq23), U = qφ0 p E 1 6 Q ij iej,

3 2 A α = J α, m = 1 d 3 r r J 2 = I 2 B m = r r ( m r), r 5 eħ µe = ge 2me, r d l, U = ( µ N µe) r 3 3( µ N r)( µe r) r 5 8π 3 ( µ N µe)δ 3 ( r) e ( µ N L) mr 3, T00 = 1 ( E 8π 2 + B 2) = a2 i + b2 i 8π T0i = ϵijk = ˆki EjBk = a 2 cos2 ( k r ωt), a 2 cos2 ( k r ωt), T ij = T i j = 1 8π (δ ij(e 2 + B 2 ) 2EiEj 2BiBj). = 1 { } a 2 δ ij aiaj bibj cos 2 ( k r ωt), 1 v ωs = ω 1 + v, (T M) Ez = ψ(x, y)e iωt+ikzz, 2 t ψ = (ω 2 k 2 z)ψ, Ψ = 0, E t(x, y) = B t(x, y) = ikz (ω 2 k z) 2 e iωt+ikzz tψ(x, y), ( ) ω ẑ Et, kz (T E) Bz = ψ(x, y)e iωt+ikzz, (ˆn t)ψ(x, y) = 0, B t(x, y) = E t(x, y) = ikz (ω 2 k z) 2 e iωt+ikzz tψ(x, y), ( ) ω kz ẑ Bt. A α (x) = E = e { B = ˆn E. d 4 x 1 x x J α (x )δ(x0 x 0 x x ), ˆn [(ˆn β) β] (1 β ˆn) 3 x }, P = 2e2 3c β 2 (Non.Rel.), dp dω = e 2 (1 β ˆn) 6 (ˆn β) β) 2, P = 2 3c e2 γ 6 [ β2 β β 2 ], dp dω = e 2 (1 β cos θ) 5 β 2 sin 2 θ (linear), dp P = 2e2 β 2 3c γ 6 (linear), dω = e 2 (1 βnβ) 5 β 2 ( (1 βnβ) 2 (1 β 2 )n 2 r ) (circular), P = 2 3c e2 β2 γ 4 (circular). dp dω = 1 8π ω4 ˆn p 2, P = ω4 3 p 2, (Thomson) σ = 8πe4 3m 2, λ = ħω λ m (1 cos θ s). electron ± muon ± proton ± neutron ±

4 LONG ANWER ECTION 1. (15 pts) Consider an infinitely long thin cylindrical shell of radius R oriented along the z axis. The shell has a surface charge density, σ = σ 0 cos ϕ. Find the electric potential at all positions as a function of the transverse radius r = x 2 + y 2 and the azimuthal angleϕ = tan 1 (y/x).

5 Extra workspace for #1

6 -Q a +Q a -Q +Q 2. Consider a set of four charges: +Q at x = a, y = a, z = 0, +Q at x = a, y = a, z = 0, Q at x = a, y = a, z = 0, Q at x = a, y = a, z = 0. (a) (5 pts) For large distances r, the electric potential can be written as Φ( r) = F (θ, ϕ)/r n. What is n? (b) (10 pts) Find F (θ, ϕ), where θ and ϕ are spherical coordinates (defined around the z axis).

7 Extra workspace for #2

8 3. An antenna is designed by circulating a current around a circular loop of radius R with its axis along the z direction. The current has the form I(ϕ, t) = I 0 cos(ωt ϕ), where ϕ denotes is the azimuthal angle of a point on the loop. (a) (5 pts) Find the charge per unit length, λ(ϕ, t). (b) (10 pts) Find the radiated power. (Use dipole approximation)

9 Extra work space for #3

10 4. A rectangular wave guide has transverse dimensions, 0 < x < a and 0 < y < a. For a transverse electric (TE) wave moving along the z axis with wave number k z. (a) (5 pts) Find the frequency of the propagating wave. Choose the solutions with the fewest nodes in the transverse wave function. (b) (10 pts) Find the magnetic field B(x, y, z, t) for this solution. (c) (5 pts) What is the group velocity of the wave?

11 Extra work space for #4

12 HORT ANWER ECTION 5. (3 pts each) Light is emitted from a distant source from early in the universe. Choose >, < or = for each answer than the frequency of the light mea- (a) The initial frequency of the source is sured by a present-day observer. (b) If an observer moves toward the source, the observed frequency will be the frequency measured by a static observer. than 6. (4 pts) In terms of M p /m e (mass of proton to mass of electron) calculate the ratio of the radiative powers P e /P p emitted for a very high-energy circular accelerator of a given radius R that features either electron or proton beams of the same energy and same currents. Note: Magnetic fields would be quite different to hold particles to the same energy and radius. 7. (3 pts each) ally lowpoke measures two events that both occur right in front of her nose separated by a time τ = 1.0 second. Roberto Rapido travels by in his space ship at some speed v. (Circle the correct answers) (a) The difference in the times of the two events Roberto measures, t, will always be positive may be positive or negative depending on v. (b) The distance between the two events measured by Robert will be always < c τ greater or less than c τ depending on v. 8. You wish to solve the following problem using the method of images: A point charge Q is placed far outside a grounded conducting spherical shell of radius R. The position of the charge is x = 0, y = 0, z = A >> R. You are solving for the potential outside the shell. True or false: (4 pts) (a) The image charge is inside the sphere. (b) The potential inside the sphere is constant. (c) The magnitude of the image charge must be less than Q. 9. (5 pts) Which of the following are odd under parity? Circle the answers. (a) A (the vector potential) (b) A 0 (the electric potential) (c) E (the electric field) (d) B (the magnetic field) (e) E B (f) B 2 E 2 (g) E 2 + B 2 (h) E B (i) J A (J is the electric current density)

13 10. Consider a system with non-zero electric and magnetic fields, B = B z ẑ, E = Exˆx, E x > 0, B z > 0, E x > B z. A charged particle is placed at the origin, initially with zero momentum. For each question answer True or False. (6 pts) (a) There exists a reference frame where B = 0. (b) At large times the coordinate x ±. (c) At large times the coordinate y ±. (d) At some parts of the trajectory the particle s momentum p x would be negative. a) R b) c) r 11. (4 pts) Consider the three charge configurations shown above: (a) two point charges +Q separated by R (b) two spheres of radius r < R/2, each with charge +Q uniformly spread throughout the volume, with the centers separated by R (c) two spherical shells of radius r < R/2, each with charge +Q uniformly spread throughout the surface, with the centers separated by R. The work required to move the spheres (or points) from infinity to the separation R is labeled W a, W b and W c for each configuration. Label each statement as true or false. (a) W a > W b (b) W a > W c (c) W b > W c

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 +