Current Balance, averaged mu0 values?

[Pete_01]

Homework Statement

While using a current balance to experimentally determine mu0 (4PI*10-7), two mu0 values were found (current was reversed to find one of these values) and averaged together in order to partially compensate for the effect of the Earth's field. Why does the flipping compensate for this and how can I prove it using taylor series expansion?

Homework Equations

f(x+e) = f(x) + f'(x)e + 0(e^R)

The Attempt at a Solution

I am not sure how to begin a proof of this!

 

[NateD]

I understand that flipping the current should make the overall magnetic field inside the current balance equal to zero, but I am unsure how to prove this using a Taylor series expansion.