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Finding Angular & Linear Momentum of EM Fields

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data
    A point charge q is a distance a>R from the axis of an infinite solenoid (radius R, n turns per unit length, current I). Find the linear momentum and the angular momentum in the fields. (Put q on the x axis, with the solenoid along z; treat the solenoid as a nonconductor, so you don't need to worry about induced charges on its surface.)[Answer: Pem0qnIR2/2a; Lem=0]


    2. Relevant equations
    [itex]\vec{E}[/itex]q=q/4[itex]\pi[/itex][itex]\epsilon[/itex]0(1/[itex]\vec{r}[/itex]2)=q/4[itex]\pi[/itex][itex]\epsilon[/itex]0([itex]\vec{r}[/itex]/r3)
    [itex]\vec{B}[/itex]sol0nI[itex]\hat{z}[/itex]
    pem0([itex]\vec{E}[/itex][itex]\times[/itex][itex]\vec{B}[/itex])
    lem=r[itex]\times[/itex]pem
    Pem=∫pem d[itex]\tau[/itex]
    Lem=∫lem d[itex]\tau[/itex]

    3. The attempt at a solution
    I kind of plugged and chugged, found r2=((x-a)2+y2+z2) and [itex]\vec{r}[/itex]=(x-a)[itex]\hat{x}[/itex]+y[itex]\hat{y}[/itex]+z[itex]\hat{z}[/itex]
    Plugged in for that as well. However, I eventually got to the integrations in for the Pem and realized I don't know what my limits of integration are for the volume.
    I know the z is from -∞ to ∞, but I have no clue for x and y. Here is what my final line looks like so far (with me already integrating over z):
    Pem=[itex]\frac{-2\mu_{0}\epsilon_{0}qnI}{4\pi\epsilon_{0}}[/itex]∫[itex]\frac{(x-a)\hat{y}}{((x-a)^{2}+y^{2}}[/itex] dydx

    Any help would be appreciated. Thanks.
     
    Last edited: Feb 20, 2012
  2. jcsd
  3. Feb 21, 2012 #2
    I figured it out. I'm too lazy to type out the solution, and since I know this could potentially help someone in the future searching for it, I'll give you the basic steps:
    1. Change the integral I had in the last line to cylindrical coordinates.
    [From here it's just tricky calculus]
    2. Do the phi integral first and break it up into two parts
    3. Here's where it gets tricky. You need to define two new variables, say A and B, where A=s^2+a^2, B=-2as.
    4. Do the integrals. DO NOT do them with a calculator or other computation engine. It WILL give you wrong answer. Use an integral table or do them by hand (good luck).
    5. Finding Lem is trivial from this point on, you find lem, then do the integral. Repeat steps 1-4 with the integral you get from Lem.
     
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