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I am trying to solve the following task

Given 3 infinitely long current carrying wires, as shown on image ($I_1=I_2$ , $I_3=2I_1$) and $r=5 cm$ calculate the point between them (in the same plane; point at $I_1$ is $x_1=0cm$, point at $x_2 = 5 cm$, point at $x_3 = 10 cm$) for which magnetic field is equal to $\vec{0}$.

Task

I tried doing this by calculating magnetic field produced by each current this way: $$ \frac{\mu I_1}{2\pi r} $$ but since I don't have $r$, I cannot do that. The point where magnetic field is equal to 0 is not at the same distance from each wire so i would get something like this : $\frac{\mu I_1}{2\pi r_1}$ for the first current, $\frac{\mu I_2}{2\pi r_2}$ for the second current, $\frac{\mu I_3}{2 \pi r_3}$ for the third current. If i put them in an equation to get resulting magnetic field I can go no further from that. Direction of the first two currents is, as shown on the image, into the screen and the third one out of the screen. What have I done wrong and how to fix it? Result is 3.3 cm but there is no way I can get it.

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up vote 1 down vote accepted

You can start by choosing an origin, and then define $r_1$, $r_2$, and $r_3$ in terms of of the $x$ and $y$ coordinate of this system. Then, realize that $\vec{B}$ has two components, which should both be zero for the magnetic field to vanish. You get two equations (one per component) in two unknowns (the coordinates), which you should be able to solve. You'll get two solution, one on either side of the horizontal axis.

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