Q. A long straight wire along the $z$-axis carries a current $I$ in the negative $z$-direction. The magnetic vector field $B$ at a point having coordinate $(x, y )$ on the $z = 0$ plane is
IIT JEEIIT JEE 2002Moving Charges and Magnetism
Solution:
Magnetic field at $P$ is $B$, perpendicular to $OP$ in the direction
shown in figure.
So, $B=B\, \sin\, \theta \hat{i}-B\cos\, \theta\hat{j}$
Here, $B=\frac{\mu_0 I}{2\pi r}$
$\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$
$\therefore B=\frac{\mu_0 I}{2\pi}.\frac{1}{r^2}(y\hat{i}-x\hat{i})$
$=\frac{\mu_0 I(y\hat{i}-x\hat{j})}{2\pi(x^2+y^2}$ (as $x^2+y^2$)
