Not your everyday wire

It is stated that the wires are not ideal.which means that they have a constant resistance per unit lenght

In the 1st case when $S1$ is turned on the current after reaching point $A$ finds two paths with different resistance.. If path $AB$ has a resistance $R$ then path $ACB$ has a resistance $3R$ as it is 3 times longer than $AB$

So the current through $AB$ , $I_{AB}$ = $\frac {3}{4}I$

And the current through $ACB$ , $I_{ACB}$ = $\frac{1}{4}I$

Now the magnetic flux intensity created by the $AB$ part in the center can be given by-

$B_{AB}$ = $\int_{0}^{\frac{\pi}{2}} \frac{\mu I_{AB}}{4\pi r} d\theta$

$B_{AB}$ = $\frac{3\mu I}{32r}$

Now again the magnetic flux intensity created by the $ACB$ part can be given by-

$B_{ACB}$ = $\int_{0}^{\frac{3\pi}{2}} \frac{\mu I_{ACB}}{4\pi r} d\theta$

$B_{ACB}$ = $\frac{3\mu I}{32r}$

So it seems that $B_{AB}=B_{ACB}$ But if we use the right hand rule we find that $B_{AB}$ and $B_{ACB}$ have opposite direction

So, $B1=B_{AB}-B_{ACB}$

$B1=0$

In the same way we would find that $B2=0$

So, $B1+B2=0.00$

$\boxed{Answer: 0.00}$

The two decimal places were just there to throw you off... :3