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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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Have you tried it and gotten a difference? If so, it would help if you summarize or show your calculations. – David Z Dec 7 '11 at 19:02

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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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? – Rebel Dec 9 '11 at 7:36
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. – Jay Bigman Dec 9 '11 at 7:58
I'll give this a shot. Would you recommend do the integration in polar coordinates or just Cartesian? – Rebel Dec 9 '11 at 8:13
A is only a function of x when it comes to the infinite, straight wire. – Rebel Dec 10 '11 at 18:10
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). – Jay Bigman Dec 10 '11 at 22:54

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