Physics - Magnetism Question with Solution | TestHub

The wire carries a current $I$ in the negative $z$ -direction. We have to consider the magnetic vector field $\vec{B}$ at $(x,y)$ in the $z=0$ plane.
Magnetic field $\vec{B}$ is perpendicular to $OP$ $\therefore \vec{B}=B\sin \theta \hat{i}-B\cos \theta \hat{j}$ $$\begin{array}{l}\sin \theta =\frac{y}{r},\cos \theta =\frac{x}{r}B=\frac{{\mu }_{0}I}{2\pi r}\\ \therefore \vec{B}=\frac{{\mu }_{0}I}{2\pi r^2}(y\hat{i}-x\hat{j})\end{array}$$ or $\vec{B}=\frac{{\mu }_{0}I(y\hat{i}-x\hat{j})}{2\pi \left(x^2+y^2\right)}$