PHYS 3327 PRELIM 1. Prof. Itai Cohen, Fall 2010 Friday, 10/15/10. Name: Read all of the following information before starting the exam:

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1 PHYS 3327 PRELIM 1 Prof. Itai Cohen, Fall 2010 Friday, 10/15/10 Name: Read all of the following information before starting the exam: Put your name on the exam now. Show all work, clearly and in order, if you want to get full credit. Box or otherwise indicate your final answers. Question 4 (e) is a bonus question worth 8 points. The total exam score cannot exceed 100, but the bonus question can help you make up points lost elsewhere. Problem # Score 1 /25 2 /25 3 /25 4 /25 4e /8 Total /100 It is your responsibility to make sure that you have all of the pages! Good luck!

2 Question 1: The Borg Revisited (25 pts) Having failed to conquer the universe with their cubical space ship, the Borg have returned with an alternative design which for all intensive purposes can be considered to be an infinite half cylinder with uniform magnetization M, as shown below in cross-sectional view. As the photon torpedo engineer on the star ship Enterprise you must quickly estimate the magnetic field distribution surrounding the Borg ship. Having been on board one of their vessels you estimate that their ship is made from a paramagnetic material with permeability µ 2. a) (5pts) Find the magnitude & direction of the bound surface current on each face in terms of M and θ. 2

3 b) (10pts) Calculate the magnitude of the magnetic field just outside the half cylinder on all faces in terms of the internal field strength B. c) (10pts) In the picture above, the magnetic field lines inside the half cylinder are shown in black. On this same figure, carefully draw the magnetic field lines just outside the half cylinder near θ =0, 45, 90 degrees and on the flat face of the half cylinder. Make sure you pay attention to the angle and direction of the B field lines. Remember, the fate of the universe is in your hands (no pressure). 3

4 Question 2: Cloaked Charge Distribution (25 pts) On the third day of lectures, Prof. Cohen claimed that if you place an arbitrary charge distribution of net charge Q in a cavity inside a spherical conductor, as shown in the figure below, then the electric field outside the sphere is just E = Q r 2 e r, where r is the distance to the center of the sphere. Your TA, Dean, was suspicious of this claim, but fortunately he knew how to solve the Laplace equation! By solving the Laplace equation outside the sphere, verify, as Dean did, that Prof. Cohen s claim is correct. Make sure the logic of your answer is clear, and please box your boundary conditions. 4

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6 Question 3: Deadly Charged Spheres (25 pts) In Charlie Sheen s 1996 movie the Arrival, aliens have invaded earth armed with charged crossed rings that produce remarkably strong E fields that can suck up entire rooms (not exactly sure how this happens). Putting what you have recently learned in Physics 3327 to use, you decide to conduct a multipole expansion for the fields produced by these devices so that you know just how far to stand away when one of these suckers goes off. You assume that each weapon is comprised of two perpendicularly crossed circular rings of radius a that are painted with a uniform line charge density ρ l and ρ l as depicted below. a) (5 pts) Compute the monopole and dipole moments of the device. 6

7 b) (12 pts) Compute the quadrupole tensor for this device. Write your final answer in matrix form. 7

8 c) (8 pts) Using your results above, compute the multipole expansion of the potential to quadrupole order. 8

9 Question 4: Safety First(25 pts) It s your niece s birthday and you decide to give her an infinite pipe of outer radius a that has a uniform surface charge σ on its outer surface. Your niece decides to spin the pipe on it s axis at angular frequency ω, as shown below. This gets you wondering about the extra stresses produced by this rotation and whether your present is safe. The pipe wall has a thickness δ a and to make the calculation easier you neglect the effects of any radiation produced by the rotation of the pipe. a) (5 pts) Show that the electric and magnetic fields produced by the pipe are, in cylindrical coordinates, E = 4πσa e r, r r>a, (1) B = 4πσωa e z, c r<a. (2) 9

10 b) (10 pts) Calculate the stress energy tensors T in and T out just inside and outside the pipe in cylindrical coordinates. Please write your final answers for T in and T out in matrix form. (Hint: take advantage of the symmetry in the problem) 10

11 c) (3 pts) Are there any shear stresses produced on the pipe by the magnetic or electric field? Explain why (in one line). d) (7 pts) Calculate the outward electromagnetic pressure on the pipe. 11

12 e) (8 pts) Extra Credit: The electromagnetic pressure in the pipe induces a so-called hoop stress τ in the azimuthal direction. As the pressure increases, the hoop stress eventually reaches a critical value τ c and the pipe wall will fracture. This would be very bad. Find the critical angular speed ω c at which failure of the pipe wall occurs, in terms of τ c, a, and σ. (Hint: You may find the hoop stress by cutting the cylinder in half along its axis, and balancing the forces produced by the radial pressure and the azimuthal hoop stress. You need only find the stress to leading order in δ/a.) 12

13 Formulae σ b = n P ρ b = P (D 2 D 1 ) n =4πρ f (E 2 E 1 ) n = 0 K b c = n M J b c = M (B 2 B 1 ) n = 0 (H 2 H 1 ) n = 4π c K f D = E +4πP P = χ e E D = εe H = B 4πM M = χ m H B = µh Φ(r) = V dv ρ(r ) r r E = Φ A(r) = V dv J(r ) r r B = A Q = α q α Φ (1) = Q r p = α q α r α Φ (2) = p r r 3 Q ij = α q α (3x iαx jα r 2 α δ ij ) Φ (4) = 1 6 ij ( 3xi x j r 2 ) δ ij Q ij r 5 Φ(x, y, z) = Φ(r, φ, θ) = α,β,γ α2 +β 2 +γ 2 =0 l l=0 m= l Φ(r, φ, z) =A 0 + B 0 log r + + m,n e ±αx e ±βy e ±γz Φ(r, θ) = [ A l r l + B ] l Y m r l+1 l (θ, φ) n=1 ([ A l r l + B l r l+1 l=0 ] [ cos(nφ)+ C l r l + D l [ ] A mn J n (k m r)+b mn N n (k m r) e ±inφ e ±kmz [ A l r l + B ] l P r l+1 l (cos θ) r l+1 ] ) sin(nφ) L 0 4π F + d dt dx sin nπx L mπx sin L = L 2 δ mn Yl m (θ, ϕ)yk n (θ, ϕ)dω =δ mn δ lk V 1 4πc E B = TdV V T ij = 1 [ ] E i E j + B i B j (E 2 + B 2 )δ ij /2 4π +1 1 ρ 0 dxp l (x)p m (x) = 2 2l +1 δ lm rdrj n (k m r)j n (k l r)= ρ2 2 J 2 n+1(k m ρ)δ ml E t + S + E J f =0 13

14 1 Vectors, derivatives A (B C) = B(A C) (A B)C, A (B C) = B (C A) (φa) = φ A + A φ (φa) = φ A A φ, (A B) = (B )A + (A )B + A ( B) + B ( A) (A B) = (B )A (A )B + A( B) B( A) (A B) = B ( A) A ( B) ( A) = ( A) 2 A ψ ψ = e r r + e 1 ψ θ r θ + e 1 ψ φ r sin θ φ A = 1 r 2 r (r2 A r ) + 1 r sin θ θ (sin θa θ) + 1 A φ r sin θ φ ( 1 A = e r r sin θ θ (A φ sin θ) A ) θ 1 + e θ φ r sin θ ( 1 + e φ r r (ra θ) A ) r, θ ( r 2 ψ ) ( 1 + r r 2 sin θ ψ ) + sin θ θ θ 2 ψ = 1 r 2 r ψ ψ = e r r + e 1 ψ θ r θ + e ψ z z A = 1 r r (ra r) + 1 A θ r θ + A z ( z 1 A z A = e r r θ A ) θ z 2 ψ = 1 ( r ψ ) ψ r r r r 2 θ ψ z 2 ( Ar + e θ z A z r ( Ar φ sin θ r (ra φ) 1 2 ψ r 2 sin 2 θ φ 2 ) + e z ( 1 r r (ra θ) 1 r ) ) A r θ

15 2 Special functions 1 1 P l (x) 2 dx = 2 2l + 1, P 0(x) = 1, P 1 (x) = x, P 2 (x) = (3x 2 1)/2 1 Y0 0 (θ, φ) = 4π, Y 1 0 (θ, φ) = 5 Y 0 2 (θ, φ) = Y ±2 2 (θ, φ) = 3 ±1 3 cos θ, Y 1 (θ, φ) = 4π 8π 15 16π (3 cos2 θ 1), Y 2 ±1 (θ, φ) = cos θ sin θe±iφ 8π 15 32π sin2 θe ±2iφ sin θe±iφ a 0 J n (kr) 2 rdr = a2 2 J n+1(ka) 2, d du [u2 J 0 (u) 2 + u 2 J 1 (u) 2 ] = 2uJ 0 (u) 2, d du J 0(u) = J 1 (u), d du [uj 1(u)] = uj 0 (u) d du [J 0(u) 2 + J 1 (u) 2 ] = 2J 1 (u) 2 /u J n (z) 2 cos[z (nπ/2) π/4], πz N n(z) 2 sin[z (nπ/2) π/4] πz