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I am not sure that I understand how to argue using symmetry arguments that the magnetic field lies in a certain direction. Say we are given a circular loop of radius $R$ with a steady current $I$ flowing in it and that we are required to find the magnetic field $B$ at a point $P$ directly above the centre of the loop at a distance $z.$ Then in this case how can I argue using symmetry that the magnetic field must lie along the $z-$ axis?

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So there you are in the $z$-axis, directly above the current loop. If you do a rotation about the $z$-axis, the loop looks identical. There is no preferred azimuthal angle. Hence, you assume the magnetic field produced by the current has no preferred azimuthal angle (is preserved by z-rotations). That only happens if

$$\vec{B} \propto \hat{z}$$

In contrast, an $x$-axis rotation completely changes how the loop appears in your coordinates so you can expect the solution to not be invariant under x-rotations (same for $y$ and any linear combination thereof).

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