Circular magnetic field around current carrying wire

Question

We know that a current carrying wire produce a magnetic field around it.But my question is why those magnetic lines are circles. How it can be like that?

Answer 1

You can derive it directly from the Biot Savart law. Let's say we have an infinite wire with a uniform current running up the $\mathbf{\hat{z}}$-axis, because of the vertical translation symmetry we can take our observation point to lie on the x axis some perpendicular distance $\rho$ from the vertical , and so $$\mathbf{B}=\frac{\mu I_0}{4\pi}\int\frac{\mathbf{\hat{r}}\times d\mathbf{z}}{\rho^2+z^2}.$$ The cross product equals $\mathbf{\hat{\phi}}\sin\theta$, where $\theta$ is the angle between the source and the observation point. We then have $$\mathbf{B}=\frac{\mu I_0}{4\pi}\int\frac{\rho \ dz}{(\rho^2+z^2)^{3/2}}\hat{\phi}.$$ $$\mathbf{B}=\frac{\mu I_0}{4\pi}\frac{2}{\rho}=\frac{\mu I_0}{2\pi\rho}$$