I am been trying to make sense of my professor's lecture notes on where he talks about line charges; in general, I am lost when it comes to the symmetry argument in the case that $E_\phi=0$ on an infinitely long line charge that is constant for $\lambda$, which could or could not be counterclockwise. At the same time, he talks about using the same symmetry argument which he claims that $B_\phi=0$ on an infinitely long wire with a constant current $I$ does not apply to the claim prior.

I know that $I$ and $B_\phi$ are not zero, but I am unsure how that would relate.


For the infinite line charge, there is nothing to differentiate clockwise from counter-clockwise around the line charge. For example, if the line is on the x-axis, then the distribution looks the same if I look from the direction of the positive axis as opposed to if I look from the negative axis. Therefore, there can be no polar component of the electric field. (We can also use symmetry arguments to say there is no field component along the wire, and thus we are left with only a radial component).

This breaks down for the infinite line current. Now it does matter which direction I look from. In one way I'll see current coming towards me, in the other I see it moving away from me. Therefore, we can now define an unambiguous direction that is in line with the current, and we can say if the field will move clockwise or counter-clockwise around this direction, i.e. using the right hand rule. (We can also use symmetry arguments to say there is no field component along the wire as well as no radial component (we want a divergence-less field here) to show the field can only have a polar component).

In summary, the two scenarios do not have the same symmetry. The direction of the current modifies the symmetry present in just the line charge.


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