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Homework No. 03 (Spring 2015) PHYS 520B: Electromagnetic Theory Due date: Monday, 2015 Feb 20, 4.30pm 1. (20 points.) The solution to the Maxwell equations for the case of magnetostatics was found in terms of the vector potential A to be Z µ0 J(r′ ) A(r) = d3 r ′ . (1) 4π |r − r′ |3 (a) Verify that the above solution satisfies the Coulomb gauge condition, that is, it satisfies ∇ · A = 0. (2) (b) Further, verify that the magnetic field is the curl of the vector potential and can be expressed in the form Z µ0 r − r′ B(r) = ∇ × A(r) = d3 r ′ J(r′ ) × . (3) 4π |r − r′ |3 2. (20 points.) The solution to the Maxwell equations for the case of magnetostatics was found to be Z µ0 r − r′ B(r) = d3 r ′ J(r′ ) × . (4) 4π |r − r′ |3 Verify that the above solution satisfies magnetostatics equations, that is, it satisfies ∇·B= 0 (5) ∇ × B = µ0 J. (6) and 3. (50 points.) (Based on Problem 5.8, Griffiths 4th edition.) The magnetic field at position r = (x, y, z) due to a finite wire segment of length 2L carrying a steady current I, with the caveat that it is unrealistic (why?), placed on the z-axis with its end points at (0, 0, L) and (0, 0, −L), is # " z−L 1 z+L µ0 I p p , (7) −p B(r) = φ̂ 4π x2 + y 2 x2 + y 2 + (z + L)2 x2 + y 2 + (z − L)2 p where φ̂ = (− sin φ î + cos φ ĵ) = (−y î + x ĵ)/ x2 + y 2 . 1 (a) Show that by taking the limit L → ∞ we obtain the magnetic field near a long straight wire carrying a steady current I, B(r) = φ̂ where ρ = p µ0 I , 2πρ (8) x2 + y 2 is the perpendicular distance from the wire. (b) Show that the magnetic field on a line bisecting the wire segment is given by B(r) = φ̂ µ0 I L p . 2πρ ρ2 + L2 (9) (c) Find the magnetic field at the center of a square loop, which carries a steady current I. Let p 2L be the length of a side, ρ be the distance from center to side, and R = ρ2 + L2 be the distance from center to a corner. (Caution: Notation differs from Griffiths.) You should obtain B= µ0 I 4 π tan . 2R π 4 (10) (d) Show that the magnetic field at the center of a regular n-sided polygon, carrying a steady current I is π µ0 I n tan , (11) B= 2R π n where R is the distance from center to a corner of the polygon. (e) Show that the magnetic field at the center of a circular loop of radius R, B= µ0 I , 2R (12) is obtained in the limit n → ∞. 4. (40 points.) (Refer Schwinger et al. problem 26.1 and the article in Ref. [1].) A simple model of a metal describes the electrons in it using Newton’s law, m dx d2 x + mγ + mω02 x = eE. 2 dt dt (13) Here the first term involves the acceleration of electron, ω0 -term binds the electron to the atoms, while γ-term damps the motion. Conductivity in typical metals is dominated by the damping term, thus mγv = eE. (14) The current density j for (constant) density nf of conduction electrons is j = nf ev. 2 (15) In conjunction we have j= nf e2 E = σE, mγ (16) where σ is the static conductivity. In 1935 Fritz London and Heinz London proposed that the current density js in a superconductor is described by the acceleration term dv m = eE, (17) dt which leads to London’s “acceleration equation” µ0 nf e2 1 djs = µ0 E = 2 E. dt m λL In terms of potentials we obtain the London equation 1 µ0 js + 2 A = ∇χ, λL (18) (19) where A is the vector potential and χ allows for a choice of gauge. (a) Using London’s equation show that a superconductor is characterized by the equations ∂js 1 µ0 = 2 E, (20) ∂t λL 1 (21) µ0 ∇ × js = − 2 B. λL (Hint: Choose the scalar potential φ = −λ2L ∂χ/∂t.) (b) Show that the magnetic field satisfies the equation 1 1 ∂2 2 ∇ − 2 2 B = 2 B. c ∂t λL (22) (c) For the static case, ∂B/∂t = 0, show that ∇2 B = 1 B, λ2L (23) which implies the Meissner effect, that a uniform magnetic field cannot exist inside a superconductor. (d) In this static limit and planar geometry, it implies − λx B = B0 e L , where λL is called the London penetration depth, mε0 2 c . λ2L = nf e2 (24) (25) Calculate the penetration depth for nf ∼ 6 × 1028 /m3 (electron number density for gold) and show that it is of the order of tens of nanometers. 3 References [1] F. London and H. London. The electromagnetic equations of the supraconductor. Proc. R. Soc. London A: Mathematical, Physical and Engineering Sciences, 149(866):71–88, 1935. 4