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

[sagigever]

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?

 

[Delta2]

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.

 

[sagigever]

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?

 

[Delta2]

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.

 

[Delta2]

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
$$\mathbf{E}=\frac{\lambda}{2\pi\epsilon_0\rho}\hat\rho$$.