How do I find the magnetic field at point $b$, very far from $a$? I know the magnetic field due to 1 current-carrying wire is $$B = \frac{\mu_0 i}{2\pi R}$$ So, does that mean the magnetic field at point $b$ is $$B = B_{top\,wire} + B_{bottom\, wire}=\frac{\mu_0 i}{2\pi R}+\frac{\mu_0 i}{2\pi R}=\frac{\mu_0 i}{\pi R}?$$ I have a mate telling me that it should be $$B = B_{top\,wire} + B_{bottom\, wire}=\frac{\mu_0 i}{4\pi R}+\frac{\mu_0 i}{4\pi R}=\frac{\mu_0 i}{2\pi R}$$ Who is correct? Him or I?
Magnetic fields are linear, which means that if you have several sources (electric currents, permanent magnets, charged particle beams, etc.) then you just add up the field from each source individually--while remembering that the field is a vector quantity.
Since point $b$ is far away from the U-turn, you can treat this setup like two separate wires carrying the same current $i$. Each wire contributes $\mu_0 i/2\pi R$, so two wires at the same distance from the point would either have twice that or zero, depending on the current direction (I'll leave it to you as an exercise to determine which).
I can't imagine where your mate got the factor of $1/2$. If he was right, then the second part of the wire makes no difference since the field at point $b$ would be the same as from one wire, which would be odd. Adding a source of magnetic field should change the field at nearby points.