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Sect 5.9: Precession of Systems of Charges in a Magnetic Field A brief discussion of something that is not rigid body motion. Uses a similar formalism. Will invoke results from E&M. Assumes you know them. Magnetic moment of a system of moving charges (summation convention): M (½)q i (r i v i ) (discrete point charges) M (½)∫dVρ e (r)(r v) (continuous distribution, charge density ρ e (r)) Angular momentum of a system of moving masses: L m i (r i v i ) (discrete point masses) L ∫dVρ m (r)(r v) (continuous distribution, mass density ρ m (r))

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Obviously, the angular momentum & the magnetic moment have similar forms. For most systems, we can show that they are proportional to each other: Define gyromagnetic ratio γ: M γL For classical systems, if the system has a uniform charge to mass ratio (q/m) we can show: γ = (q/2m) For quantum systems, with spin, γ is more complicated We leave it as a parameter in what follows. An E&M result: In the presence of an external magnetic field B, a magnetic dipole M will experience forces & torques. Can derive these from a potential of the form: V - (M B)

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E&M result: In an external B field, the torque experienced by a magnetic moment M is given by N M B Newton’s 2 nd Law (rotational version!) : The total torque on a system = the time rate of change of angular momentum (use M γL) (dL/dt) N =M B = γ(L B) = - (γB) L (1) From the rotational dynamics discussion, (1) If L is constant in length, but is rotating (precessing) in space about the direction of B with angular velocity ω, its eqtn of motion is: (dL/dt) ω L (2) Comparison of (2) with (1) In the presence of an external B field, the angular momentum L of a charged particle system precesses about B direction with frequency ω - γB

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(dL/dt) ω L (2) ω = - γB Larmor Frequency In an external B field, the angular momentum L of a charged particle system precesses uniformly about the B direction with frequency ω = - γB Clearly, since we have M γL, it is also true that (dM/dt) ω M (3) In an external B field, the magnetic moment M of a charged particle system precesses uniformly about the B direction with frequency ω = - γB Using the classical gyromagnetic ratio, we have: ω = - γ B = -(q/2m)B For electrons, q = -e & the precession is counterclockwise around the B direction. Basis for classical treatment of magnetic resonance!

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Now consider a slightly different problem: Consider a collection of moving charged particles, no restriction on their motion, but all with same ratio (q/m). Constant magnetic field B. Assume interactions depend only on interparticle distances. The Lagrangian is (from Ch. 1, summation convention) : L = (½)m i (v i ) 2 + (q/m)m i v i A i (r i ) - V(|r i -r j |) (1) A i (r i ) = vector potential E&M result: B = A. Constant field B A is of the form: A = (½)B r. Put into (1), interchange dot & cross products: L = (½)m i (v i ) 2 + (½)(q/m)B (r i m i v i ) - V(|r i -r j |) (2) Note that r i m i v i L i (angular momentum) L = (½)m i (v i ) 2 + (½)(q/m)(B L) - V(|r i -r j |) (3) Or: using γ = (q/2m) and M = γL L = (½)m i (v i ) 2 + (M B) - V(|r i -r j |) (3´)

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L = (½)m i (v i ) 2 + (½)(q/m)(B L) - V(|r i -r j |) (3) L = (½)m i (v i ) 2 + (M B) - V(|r i -r j |) (3´) Using ω = - γB = -(q/2m)B, it is convenient to rewrite the (½)(q/m)(B L) = (M B) term as (summation convention): (½)(q/m)(B L) = - ω L i = - ω (r i m i v i ) (4) Now, rewrite (3) in terms of coordinates relative to a primed axes set, which has a common origin with the unprimed set, but is rotating uniformly about B with angular velocity ω. –The distances are unchanged: (|r i ´- r j ´|) = (|r i - r j |) –However, the velocities transform as they do in the rigid body rotation problems as: v i = v i ´ + (ω r j )

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(½)m i (v i ) 2 = (½)m i [v i ´ + (ω r j )] 2 = (½)m i (v i ´) 2 + m i (v i ´) (ω r j ) + (½)m i (ω r j ) (ω r j ) (a) and: - ω (r i m i v i ) = - ω (r i m i v i ´) - ω [r i m i (ω r j )] (b) In the Lagrangian L, the terms linear in ω & v i ´ cancel out. Work on the quadratic terms in ω : Gathered together, they can be written - (½)ω I ω - (½)I (ω ) 2. I Inertia tensor about the direction of ω I Moment of Inertia about the direction of ω So, the Lagrangian is: L = (½)m i (v i ´) 2 - V(|r i -r j |) - (½)I (ω ) 2 (5)

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So, the Lagrangian for a system of charged particles in a constant external B field is: L = (½)m i (v i ´) 2 - V(|r i -r j |) - (½)I (ω ) 2 (5) Note! There are no linear terms in magnetic the field B (or the frequency ω ). In the rotating frame, the lowest order term in the magnetic field is quadratic. See text, where it is argued that the 3 rd term in (5) is << 1 st 2 terms. Larmor’s Theorem: To 1 st order in B, the effect of a constant B field on a classical system is to superimpose on the motion a uniform precession about the B direction at angular velocity ω

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