Today in Physics 217: magnetic multipoles
Multipole expansion of the magnetic vector potential z Magnetic dipoles θ B Magnetic field from a w magnetic dipole m Torque on a magnetic y dipole in uniform B θ Force and energy and x magnetic dipoles
13 November 2002 Physics 217, Fall 2002 1 Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. Its vector potential at point r is r Id r =−rr′ Ar()= . θ c ∫ r
Just as we did for V, we can expand 1 r I r’ in a power series and use the series as an approximation scheme: d ∞ n 11 r′ = ∑ Pn ()cosθ r rrn=0 (see lecture notes for 21 October 2002 for derivation).
13 November 2002 Physics 217, Fall 2002 2 Multipole expansion of the magnetic vector potential (continued) Put this series into the expression for A: I 11 Ar()=+ drd ′cosθ Monopole, dipole, cr ∫∫r2 1322 1 +−rd′ cos θ quadrupole, r3 ∫ 22 1533 3 +−rd′ cosθθ cos octupole r4 ∫ 22 +…
Of special note in this expression:
13 November 2002 Physics 217, Fall 2002 3 Multipole expansion of the magnetic vector potential (continued) The monopole term is zero, since ∫ d = 0. This isn’t surprising, since “no magnetic monopoles” is built into the Biot-Savart law, from which we obtained A. For points far away from the loop compared to its size, we obtain a good approximation for A by using just the first (or first two) nonvanishing terms. (For points closer by, one would need more terms for the same accuracy.) This is, of course, the same useful behaviour we saw in the multipole expansion of V.
13 November 2002 Physics 217, Fall 2002 4 Magnetic dipoles
II Ardipole ()==⋅rd′′cosθ () rrˆ d cr22 ∫∫ cr Note that ddd()rrˆˆˆ⋅=⋅+⋅′′ r r r( ′′ r rr )() ′ r ′ () ( ) () dddrrˆˆˆ⋅=⋅+⋅′′ r r r ′′ r rr ′ r ′ ∫∫∫() but ∫ d ()rrˆ ⋅=′′ r 0 , so () ∫∫rrrˆˆ⋅=−⋅dd′′ rrr ′ ′. Also, rrrˆˆˆˆ××= ∫∫′dd ′ rrr ′()() ⋅− ′ ∫ rrr ⋅ ′′ d =−⋅2. ∫ rrr ′′ d ()
13 November 2002 Physics 217, Fall 2002 5 Magnetic dipoles (continued) 1 Thus ()rrˆˆˆ⋅=⋅=−××′′dd r rr ′ () r r ′ d ∫∫2 ∫ 1 I so Arrdipole =−ˆ ×′ ×d 2 cr2 ∫ rmˆˆ×× mr ≡− = , rr22 I where mr≡×′ d . 2c ∫
pr⋅ ˆ (Compare to Vddipole ==,.)prρτ′′ r2 ∫
13 November 2002 Physics 217, Fall 2002 6 Magnetic dipoles (continued)
To see what this means in terms of geometry of a πα− current loop and its dipole d α moment, consider the triangle r’ formed by r′ and d : 11 1 r′′×=drdrd sinαπα = ′ sin () − 22 2 = area of triangle. 1 m So ra′′×=dd , and 2 II maa== d ′ for plane loops. a cc∫ I
13 November 2002 Physics 217, Fall 2002 7 Magnetic field from a magnetic dipole
For a dipole with mz = m ˆ , we have, in spherical coordinates, mr× ˆ msinθ Adipole == φˆ , rr22 rˆ ∂∂θˆ so BA=×— =sinθ ArAφφ − dipole dipole rrrsinθθ∂∂ mmrˆ ∂∂sinθθˆ 1 =−sin2 θ r3 sinθ ∂∂θ rrr mmrˆ sinθθˆ =+2sinθθ cos rr33sinθ 2cosmmθθ sinˆ 1 =+=⋅−rˆˆˆθ 3.()mrr m rrr333
13 November 2002 Physics 217, Fall 2002 8 Magnetic field from a magnetic dipole (continued)
That is to say, B from magnetic dipoles looks exactly like E from electric dipoles. We can thus work out forces, torques, energies, and even a lot about magnetostatics in magnetically-polarizable matter in strict analogy with electrostatics. The real- dipole paradigms:
+q a s Planar loop -q I p = qs m = Ia/c (Ia in MKS)
13 November 2002 Physics 217, Fall 2002 9 Torque on a magnetic dipole in uniform B
z z
θ B m B IwB w m F = c θ View down x θ y y θ IwB F = x c
Iw NF== sinθθθ B sin = mB sin c ⇒=×NmB()cf. NpE =× .
13 November 2002 Physics 217, Fall 2002 10 Force and energy and magnetic dipoles
Thus the following should be no surprise: FmB=⋅—() =⋅()mB— if no currents exist at dipole's position. (cf. Fp=⋅()— E) WW=−mB ⋅(cf. =− pE ⋅ )
Magnetic dipoles thus tend to align in magnetic fields in the same way that electric dipoles align in electric fields.
13 November 2002 Physics 217, Fall 2002 11