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Finding the gyromagnetic ratio of an axially symmetric body

  1. Feb 23, 2014 #1
    1. The problem statement, all variables and given/known data

    So, we can presumably write that m = g L, where L is the angular momentum, g the ratio wanted, and m magnetic dipole moment of an axially symmetric body. Total mass is M, total charge Q, mass density [itex]\rho_m(r)=\frac{M}{Q}\rho_e(r)[/itex], where [itex]\rho_e(r)[/itex] is charge density.

    2. Relevant equations

    Moment of inertia can also be written [itex]I_\omega=\int \rho_m d^2 d\tau[/itex] where d is the distance from the axis of symmetry.

    3. The attempt at a solution

    I guess the dipole moment is in the direction of the axis of the symmetry, as is the angular velocity and it can be written:
    m = g I[itex]_\omega [/itex] ω

    RHS:
    = g ω [itex]\hat{z}[/itex] I[itex]_\omega [/itex]

    LHS:
    m = I [itex] \int d\vec{a} [/itex]

    Here is where I guess I'm having conceptual problems. I is the total current, part of which should be a current flowing through a differential circular loop inside the body, at distance d, that is to say:

    I = [itex]\int \left| v \right| \rho_e d\tau [/itex] , where [itex]\left|v\right|[/itex] is the module of radial speed of a differential volume element (at distance d) that can be written: [itex]\left|v\right|=\left|\omega\right|\left|r\right|[/itex]sin[itex]\theta [/itex]= ωd. [itex] \int d\vec{a} [/itex] of a differential loop in question is simply [itex] d^2 \pi[/itex]. So, im getting:

    g ω [itex]\int \rho_m d^2 d\tau[/itex] =ω [itex]\int d^3 \pi \frac{M}{Q} \rho_m d\tau [/itex]

    Where did I go wrong, how to find g?
     
  2. jcsd
  3. Feb 24, 2014 #2
    anyone? :( the integrals are supposed to annihilate each other leaving me with a factor ~M/Q
     
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