# Magnetic field generated by circular current carrying loop

I'm trying to evalueate the magnetic field generated by circular current carrying loop. My manual says that for the symmetry of the sytem only the magnetic field along the $$z$$ axis (which is the symmetry axis of the conductor) is not zero. I don't understand the reason. It seems that if I consider a point $$P$$ on the $$xy$$ plane, $$\mathbf{H}$$ should not vanish following my calculations. The elementary magnetic field generated by a infinitesimal segment is:

$$d\mathbf{H} = I\frac{ x \hat{r}}{4\pi r^2}d\mathbf{s}$$

• as you said "the magnetic field along the z axis( which is the symmetry axis of the conductor) is not zero.", i.e, the field is not zero in the middle per your calculation. Note that in your formula "$d\vec s \hat r$" refers to a vector product and when you are at the center the vectors are orthogonal. Commented Oct 4, 2020 at 11:24
• I think the manual is calculating the magnetic field only on the $z$ axis ignoring points outside it. For points along $z$ I think it is saying, there are no components of the magnetic field along $x$ and $y$. $B_y = B_x = 0$. Commented Oct 4, 2020 at 11:29
• indeed, and it follows from the cylindrical symmetry about the z axis. Commented Oct 4, 2020 at 11:32
• why Does it limit the treatment to a particular axis and does not consider as well points on $xy$ plane for example? Commented Oct 4, 2020 at 11:34
• everywhere else off axis the result of integration is a nasty elliptic integral not expressible in elementary functions Commented Oct 4, 2020 at 11:37