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Magnetic Vector Potential of Coil

  1. Feb 22, 2014 #1

    VVS

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    Hi

    Basically I want to examine the effect of a magnetic vector potential created by a coil on the spin of an electron in a Coulomb potential.
    The Hamiltonian of a charged particle in a Vector Potential is well known.
    But I have a problem in calculating the Magnetic Vector Potential of a finite lenght Coil.

    1. The problem statement, all variables and given/known data

    The equation for a magnetic vector potential is given by.

    [itex]\vec{A}(\vec{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Re^{3}}\frac{\vec{J}(\vec{r}',t)}{\left|\vec{r}-\vec{r}'\right|}d^{3}\vec{r}'[/itex]

    2. Relevant equations
    The vector equation in cylindrical coordinates for a coil is

    [itex]\vec{r}'=\hat{i}\rho_{0} cos(\vartheta)+\hat{j}\rho_{0} sin(\vartheta)+\hat{k}\frac{\vartheta}{2\pi}[/itex]

    Therefore the equation for the Current Density is

    [itex]\vec{J}(\vec{r}',t)=(-\hat{i}\rho_{0} sin(\vartheta)+\hat{j}\rho_{0} cos(\vartheta)+\hat{k}\frac{1}{2\pi})\delta (\rho-\rho_{0})[/itex]

    The position of any point in space in cylindrical coordinates is given by

    [itex]\vec{r}=\hat{i}\rho cos(\vartheta)+\hat{j}\rho sin(\vartheta)+\hat{k}z[/itex]

    3. The attempt at a solution
    One can write the Volume integral in cylindrical coordinates.

    [itex]\vec{A}(\vec{r},t)=\frac{\mu_{0}}{4\pi}\int_{0}^{∞}\int_{0}^{2\pi}\int_{-h/2}^{h/2}\frac{(-\hat{i}\rho_{0} sin(\vartheta)+\hat{j}\rho_{0} cos(\vartheta)+\hat{k}\frac{1}{2\pi})\delta (\rho-\rho_{0})}{(\rho-\rho_{0})^2+(z-\frac{1}{2\pi})^2}\rho dz d\vartheta d\rho[/itex]

    And performing the integral you finally end up with only the k component.
    Which must be wrong because I know that the Magnetic Vector Potential is finite outside of the coil.

    Please help me out.
     
  2. jcsd
  3. Feb 22, 2014 #2
    Hi.
    Typically, direct integrations of this kind are very difficult and you're gonna have a hard time if you don't make any simplifications. For example, the field here is likely to get very complicated near the coil so i would certainly not expect a solution in closed form... Instead i would work with a solenoid, for which the magnetic field is easy to determine and is a good approximation as long as you don't get too close to the coil, then find a suitable vector potential from that.
    (Incidentally, in your attempt of a solution the denominator is wrong...)
     
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