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http://www.ux1.eiu.edu/~cfadd/1360/30MagSrcs/Images/FigP30.08.jpg

In calculating the magnetic field created by this current at the center point of the loop using Biot-Savart and using the vector potential will there be a difference? If so what is it and why?

Using Biot-Savart the Magnetic field is simply a superposition of fields created by straight wire and loop. But how does one calculate the vector potential at the center?

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    $\begingroup$ Have you tried it and gotten a difference? If so, it would help if you summarize or show your calculations. $\endgroup$
    – David Z
    Commented Dec 7, 2011 at 19:02

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The vector potential at point $x$ can be calculated as the integral over the wire of the vector current divided by the distance to $x$. Take the curl of the resulting expression, move the curl inside the integral sign, do a few manipulations and wala!, you've got the Biot-Savart law! They are exactly the same.

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  • $\begingroup$ I haven't been able to set up the integral to calculate the vector potential that doesn't diverge. And what do you mean move the curl inside the integral? $\endgroup$
    – Rebel
    Commented Dec 9, 2011 at 7:36
  • $\begingroup$ I see what's bothering you, the 1/r divergence. That disappears when you differentiate under the integral sign. $\partial_{x} \int dy A(x,y) = \int dy \partial_{x} A(x,y)$. A mathematician might take offense. $\endgroup$
    – Jay Bigman
    Commented Dec 9, 2011 at 7:58
  • $\begingroup$ I'll give this a shot. Would you recommend do the integration in polar coordinates or just Cartesian? $\endgroup$
    – Rebel
    Commented Dec 9, 2011 at 8:13
  • $\begingroup$ A is only a function of x when it comes to the infinite, straight wire. $\endgroup$
    – Rebel
    Commented Dec 10, 2011 at 18:10
  • $\begingroup$ I know, I was just trying to show what I mean by differentiating under the integral sign. Why don't you calculate the vector potential for the ring separately (it won't diverge). $\endgroup$
    – Jay Bigman
    Commented Dec 10, 2011 at 22:54

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