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!

Retarded potential of a wire loop

  1. Dec 15, 2012 #1
    1. The problem statement, all variables and given/known data

    This is the problem 10.10 in Griffiths' Introduction to Electrodynamics, 3e

    (See attached image)

    I am only concerned about finding the retarded vector potential [itex] \textbf{A} [/itex]

    2. Relevant equations

    [itex] \textbf{A}(r',t) = \frac{\mu_{0}}{4\pi} \int \frac{ \textbf{I}(t_{r}) dl}{r'}[/itex]
    [itex] t_{r} = t - r'/c [/itex]


    3. The attempt at a solution

    I got this far:

    [itex] \textbf{A}(r',t) = \frac{\mu_{0}k}{4\pi} \left[ t \int \frac{\textbf{dl}}{r'} + \int \frac{\textbf{dl}}{c} \right][/itex]

    Which matches the given solution. My next step would be to take the first integral and divide it into components, that is, take the line integral over the inner semi circle, the outer semi circle, and the two linear components on the x axis seperately. For the line integral over the inner circle, I would take,

    [itex] \int_{inner} \frac{\textbf{dl}}{r'} = -\hat{\theta} \int_{0}^{\pi} \frac{a d\theta}{a}[/itex]

    which should be equal to [itex] -\pi \hat{\theta} [/itex] (r' is a in this case because we want to find the potential at the centre). However, the solution says that

    [itex] \int_{inner} \frac{\textbf{dl}}{r'} = 2a\hat{x} [/itex]

    The solution given for the outer integral is similar.

    Either a) I've forgotten how to do line integrals (a distinct possibility), b) There is some physical reason for the solution given or c) the solution is incorrect.

    I'd be very much obliged if someone could let me know what is going on for the solution given, and why my approach is incorrect.
     

    Attached Files:

  2. jcsd
  3. Dec 16, 2012 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello delyle. Welcome to PF!

    In the integrals, dl is a vector. So the integration corresponds to vector addition. You should consider 2 integrals, one for adding the x-components of dl to find the x-component of A and another integral for the y-component.


    Note that [itex] \int_{inner} \frac{\textbf{dl}}{r'} = 2a\hat{x} [/itex] is dimensionally inconsistent. The left side is dimensionless while the right side has dimension of length.
     
  4. Dec 16, 2012 #3
    Hello TSny,

    Thank you for your reply. I think I understand now. As far as the dimensional inconsistency, I had made a mistake in my explanation; the solution said that [itex] \int_{inner} d\textbf{l} =2a\hat{x} [/itex], which makes more sense, dimensionally.

    So, let's see if I follow you. Explicitly, [itex] -d\textbf{l} = a\sin(\theta)d\theta\hat{x} - a\cos(\theta)d\theta\hat{y} [/itex], and so the integral [itex] \int_{inner} d\textbf{l} [/itex] can be rewritten as [itex] a\int_{0}^{\pi} \left[\sin(\theta)\hat{x} - \cos(\theta)\hat{y} \right]d\theta [/itex]. The integral over cos is 0, so we're left with [itex] \int_{inner} d\textbf{l} = 2a\hat{x} [/itex], which is consistent with the solution. Thanks a ton!
     
  5. Dec 16, 2012 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    That's essentially it. I'm not sure I see your choice of signs, but that might have to do with how you're defining the positive direction of θ. dl points in the same direction as the current. So along the arc of radius a, dl should have a positive x-component, but you wrote -dl as having a positive x-component (I think). Anyway, your result looks good. :smile:
     
  6. Dec 16, 2012 #5
    Yes, you're right. That minus sign in front of [itex] d\textbf{l} [/itex] should be removed
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Retarded potential of a wire loop
  1. Retarded potential (Replies: 7)

  2. Retarded Potentials (Replies: 4)

  3. Retarded potentials (Replies: 3)

Loading...