Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 7 Maximal score: 25 Points. 1. Jackson, Problem Points.

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1 Physics 505 Eecticity and Magnetism Fa 00 Pof. G. Raithe Pobem et 7 Maxima scoe: 5 Points. Jackson, Pobem 5. 6 Points Conside the i-th catesian component of the B-Fied, µ 0 I B(x) ˆx i ˆx i d (x x ) x x ˆx i d x x x d x x x ˆx i tokes aw { } da x x x x ˆx i { da x ( x ˆx i ) ˆx i x x x x x } +(ˆx i x ) x x x ( x x x x )ˆx i use x x aways { } da (ˆx i x ) x x x 0 { } da (ˆx i x ) x x x 0 { } { } da x x i x x da x x i x x { } da x x i x x see Eqn. afte.5 on page of textbook dω Ω(x) x i x i Thus, B i µ0i x i Ω(x), and B(x) µ 0I xω(x) q.e.d.

2 . Jackson, Pobem 5. 6 Points Conside a oop cuent with adius R aound the z-axis. The oop is centeed at ocation ẑz. Then, the magnetic fied at an obsevation point ẑz is B(0, 0, z) µ 0I d x x x x Inset x R cos φ R sin φ z and d R sin φ R cos φ 0 dφ to find B(0, 0, z) µ 0I π Rz cos φ Rz sin φ R + z 0 R dφ ẑ µ 0I R R + z Now, conside a soenoid the axis of which coincides with the z-axis. The soenoid has N windings pe ength, cuent I, and end points z and z. Then, the amount of cuent fowing acoss a ength dz is di INdz, and B(0, 0, z) ẑ µ 0dI ẑ µ 0R IN ẑ µ 0R IN ẑ µ 0IN with anges θ and θ as shown in the pobem statement. R R + z z dz z R + z z R R + z cos θ + cos θ, z z

3 . Jackson, Pobem Points a): In the Couomb gauge, A µ 0 J φ (, θ) ˆφ. Using a vaiabe sepaation method, we constuct a soution of the fom A A φ (, θ) ˆφ. (By finding the soution of that fom, it is shown that it exists.) The foowing deivatives of spheica unit vectos ae usefu: ˆφ 0 θ ˆφ 0 φˆ ˆφ sin θ φ ˆθ ˆφ cos θ φ ˆφ ˆ sin θ ˆθ cos θ φ ˆφ ˆφ. Thus, witing out (A φ (, θ) ˆφ ) µ 0 J φ (, θ) ˆφ in spheica coodinates yieds: ( + ( + ) sin θ ( ˆφ) θ sin θ θ + sin θ φ A φ (, θ) µ 0 J φ (, θ) ˆφ ) ( ) sin θ θ sin θ θ A φ (, θ) ˆφ sin θ A φ(, θ) ˆφ µ 0 J φ (, θ) ˆφ + sin θ θ sin θ θ sin A φ (, θ) µ 0 J φ (, θ) θ This is a -nd ode, inea, inhomogeneous PDE fo A φ (, θ), simia to the Poisson equation, which is sovabe. To identify the behavio inside and outside the cuent distibution, we sove the homogeneous equation by sepaation of vaiabes. Witing A φ (, θ) U() Θ(θ), it is d U() U() }{{} + d sin θ dθ sin θ d Θ(θ) dθ Θ(θ) sin θ Θ(θ) }{{} ( + ) ( + ) 0 The angua equation is the geneaized Legende diffeentia equation, d sin θ dθ sin θ d dθ sin + ( + ) Θ(θ) 0, θ which has the egua soution P (cos θ); note that P (cos θ) is ineay dependent. This finding justifies a posteioi that ( + ) with,... is a good choice fo the sepaation vaiabe. The adia equation, has the soution d ( + ) U() U() 0

4 U() A + + B. ummaizing, the inteio and exteio soutions ae found to be A φ,inteio A φ,exteio A P (cos θ) µ 0 B P (cos θ) µ 0 m P (cos θ) µ P (cos θ) q.e.d. () Thee, we aso define the mutipoe moments m and µ. Note that the { P (x),,,..} fom a compete othogona set on the inteva,. b): In anaogy with eectostatics, spheica mutipoe moments ae obtained by expanding x x in spheica hamonics. Fo azimutha cuent distibutions it is, fo an obsevation point with φ 0, A φ (, θ) µ 0 µ 0 J(, θ, φ ) ˆφ x x d x µ 0 + Y m(θ, φ 0) <,m Jφ (, θ ) ˆφ ˆφ + x x d x µ 0 Jφ (, θ ) cos φ x x d x Y m(θ, φ ) exp(iφ ) + exp( iφ ) J φ (, θ )d x Upon integation ove φ, ony m ± give non-zeo contibutions. Fo each, the m and m tems ae equa; to show this, use the fact that φ 0, and Eqs..5 and.5 of the textbook. Thus, A φ (, θ) µ 0,m < + µ 0,m ( )! ( + )! P (cos θ) ( )! ( + )! P (cos θ ) J φ(, θ )d x < ( + ) P (cos θ) + P (cos θ )J φ (, θ )d x () Fo the inteio egion, < and, and A φ,inteio (, θ) µ 0 { ( + ) + P (cos θ )J φ (, θ )d x } P (cos θ) Compaison with Eq. shows that m ( + ) + P (cos θ )J φ (, θ )d x q.e.d. 4

5 imiay, fo the exteio egion, and <, and A φ,inteio (, θ) µ 0 { ( + ) P (cos θ )J φ (, θ )d x } + P (cos θ) Compaison with Eq. shows that µ ( + ) P (cos θ )J φ (, θ )d x q.e.d. 5

6 4. Jackson, Pobem 5. 7 Points Thee is an azimutha suface cuent K(θ ) ˆφ σ sin θ aω. The coesponding thee-dimensiona cuent density is j(, θ ) ˆφ K(θ )δ( a) ˆφ σ sin θ aωδ( a) ˆφ J φ (, θ ). Using Eq. of the pevious pobem and P m (x)p m (x)dx (+m)! + ( m)! δ, and P sin θ, it is A φ (, θ) µ 0 µ 0 µ 0,m,m,m ( + ) P (cos θ) < ( + ) P (cos θ) µ 0σa ω P (cos θ) < 4 4 µ 0σa ω + < + ( + ) P (cos θ) πσa ω < + sin θ < P (cos θ )J φ (, θ )d x P (cos θ )σ sin θ δ( a) a ω d cos θ dφ P (x)p (x)dx Thus, it is outside the sphee and inside A exteio (, θ) ˆφ µ 0σa 4 ω A inteio (, θ) ˆφ µ 0σaω sin θ sin θ Using that fo azimutha A it is B A ˆ sin θ θ sin θa φ ˆθ A φ it is found: B exteio (, θ) µ 0σa 4 ω ˆ cos θ + sin θ ˆθ, which is the fied of a magnetic dipoe m ẑ σa4 ω, and B inteio (, θ) µ 0σaω ˆ cos θ ˆθ sin θ, which is a homogeneous magnetic fied in z-diection. 6