Properties of symmetric magnetic field around ##Z## axis (cylinder)

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Summary:

in cylinder coordinates, I am trying to understand but without a succes why symmetric magnetic field around ##Z## axis make that ##\hat \phi## magnetic field is zero
I am trying to understand but without a succes why symmetric magnetic field around ##Z## axis make that ##\hat \phi## magnetic field is zero
I can't understand why it physically happens and also how can I derive it mathematically?
What does the word symmetric means when talking about magnetic field?
 

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Delta2
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Cylindrical symmetry around z-axis essentially means that the field depends only on ##\rho## (and possibly ##z## but not ##\phi##). It doesn't necessarily means that it is in the ##\hat\phi## direction or that it is zero.
 
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Cylindrical symmetry around z-axis essentially means that the field depends only on ##\rho## (and possibly ##z## but not ##\phi##). It doesn't necessarily means that it is in the ##\hat\phi## direction or that it is zero.
can you give me example when it not zero in ##\hat \phi##?

ohh so you mean that it can have component in the ##\phi## direction, but the derivative with respect to ##\phi## will always be zero?
 
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Delta2
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can you give me example when it not zero in ##\hat \phi##?

ohh so you mean that it can have component in the ##\phi## direction, but the derivative with respect to ##\phi## will always be zero?
yes. An example where we have cylindrical symmetry and the field is not zero (however its in the ##\hat\phi## direction) is the magnetic field of an infinitely long thin wire that lies at the z-axis and carries current I. Then the field is $$\mathbf{B}=\frac{\mu_0}{2\pi r}I\hat\phi$$. As you can see it depends only on ##r## (not on ##\phi## or ##z##) and it is not zero.
 
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Delta2
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An example of a cylindrical symmetric field that is in the ##\hat\rho## direction is that of the electric field of an infinitely long thin charge density ##\lambda## that lies again in the z-axis. The Electric field is given by
$$\mathbf{E}=\frac{\lambda}{2\pi\epsilon_0\rho}\hat\rho$$.
 

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