Average Magnetic Field Between 2 Conducting Rods

[Gbl911]

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

 

[rude man]

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.