I Average Magnetic Field Between 2 Conducting Rods

AI Thread Summary
The discussion focuses on calculating the average magnetic field between two conducting copper rods for a railgun project. The formula provided for the magnetic field at a point between the rods involves parameters such as current and distance from the rods. The user seeks assistance in deriving an expression for the average magnetic field across the gap, acknowledging the need for integration. A solution is suggested that incorporates the finite radius of the wires to avoid issues as the distance approaches zero. Understanding the derivation and application of this average magnetic field formula is essential for the project.
Gbl911
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I am building small, simple version of a railgun using 2 copper bars and a couple of neodymium magnets to increase the magnetic field. I have also been trying to mathematically describe the magnetic field created by the conducting rods themselves. I am coming across some trouble when trying to derive an expression for the average magnetic field across the entire gap and would like some help.

You can write the magnetic field at anyone point in between the two bars as

(u*I)/(2*pi) * (1/r + 1/(d-r))
where u is mu, d is distance between the bars, I is the current, and r is the distance from the center of one bar to the point of interest.
I assume that I need to compute some form of integral but I'm not sure exactly what to integrate over and how to obtain the average field strength from that.
I found a website that details how they went about it but I still am not sure on why the way they did it works. Here is the link.
https://military-history.fandom.com/wiki/Railgun
 
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You will notice that there is a problem as ## r \rightarrow 0 ##. So you need to take the finite radius of your wires into account.
If you do that the average is ## B_{avg} = ~(2\int_a^{d-a} \mu_0 I/2\pi r~ dr)/(d-2a) ## where ## a ## is the wire radius.
Not sure why you want this but there you go.
 
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