9.2.6. Long straight wires with current intersect at right angles. Determine the magnetic field induction at a point with coordinates $x$ and $y$ if the coordinate axes are the wires and the current in the wires is $I$.
Solution
Direction of magnetic induction created by current in wires
We find the total vector of magnetic induction as the sum of the vectors of magnetic induction created by each of the vectors $$\vec{B} = \vec{B_1} + \vec{B_2}$$ Using the right-hand rule, we can determine the direction of the magnetic induction lines and notice that $\vec{B_1}$ and $\vec{B_2}$ are directed perpendicular to the $xy$ plane, which means they are co-directed $$\boxed{B=B_1+B_2}\tag{1}$$ We will find the magnetic induction of each of them as the induction of an infinite single-horned wire. $$B_1 = \frac{\mu_0 I}{2\pi} \cdot \frac{1}{x};\quad B_2 = \frac{\mu_0 I}{2\pi} \cdot \frac{1}{y}\tag{2}$$ We substitute the obtained expression $(2)$ into $(1)$$$\boxed{B = {B_1} +{B_2} = \frac{\mu_0 I}{2\pi}\cdot \left(\frac{1}{x}+\frac{1}{y}\right)}$$
