1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Magnetic Vector Potential of Coil

  1. Feb 22, 2014 #1


    User Avatar


    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.


    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
    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...)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted