Infinite wire: Symmetry of magnetic field caused by a cylindrically symmetric current

Question

Recall how we find the Electric field of spherically symmetric charge distribution, cylindrically symmetric charge distributions and also charge distributions with planar symmetry (infinite flat sheet), by Gauss's Law and arguements based on symmetry alone. This make me think that we can do the same in magnetism by using Ampere's law and symmetry alone. But I failed. So I would like to know if I have used the wrong method of arguments.

So take an infinite straight long current carrying wire as the simplest example. WLOG, let it be lying on the $z$ axis with current flowing in the positive $z$ direction. So using cylindrical coordinate system $$\vec{B}(\rho,\phi,z)=B_\rho(\rho,\phi,z)\hat{\rho}+B_{\phi}(\rho,\phi,z)\hat{\phi}+B_z(\rho,\phi,z)\hat{z}$$

Now by translational symmetry along the wire, we conclude that nothing can have $z$-dependence. And by rotational symmetry about $z$ axis, nothing can have $\phi$ dependence. Hence $$\vec{B}(\rho)=B_\rho(\rho)\hat{\rho}+B_{\phi}(\rho)\hat{\phi}+B_z(\rho)\hat{z}$$

Now if you flip the current upside down, $B_{\rho}(\rho)\hat{\rho}$ is unchanged. While it should have become $-B_{\rho}(\rho)\hat{\rho}$ because superimposing the flipped current and the original current gives zero current and therefore zero field.

Hence now we have $\vec{B}(\rho)=B_{\phi}(\rho)\hat{\phi}+B_z(\rho)\hat{z}$$

Now here comes the problem: How to show that $$B_z(\rho)\hat{z}=\vec{0}$$

I saw some people using Biot-Savart Law at this point, appealing to the fact that a current in the $z$ direction produce no $B_z\hat{z}$.

So is it possible to reach the same conclusion by using symmetry and superposition principle and Ampere's law only, without asking for help from Biot-Savart Law?

Similar issues arises in deriving the field due to an infinitely long solenoid. It seems that in addition to symmetry, superposition, and Ampere's law, we also need Biot-Savart law and Gauss's law for magnetism as well. I will post a separate question soon if necessary.

Answer 1

Here is one argument:

Draw a rectangle with normal pointing in the $\phi$ direction and not enclosing the line of current. Suppose the rectangle has height $h$ and ranges from $\rho = a$ to $\rho = b$. Since $B_\rho = 0$, the line integral of $\mathbf{B}$ around this rectangle is simply $h(B_z(b)-B_z(a))$, which must be zero by Ampère's law. So $B_z$ doesn't depend on $\rho$, since $a$ and $b$ were arbitrary. If we want $\mathbf{B}$ to vanish as we move infinitely far from the wire, then $B_z$ is zero everywhere.

Incidentally, I'm not sure I understand your argument that $B_\rho$ must be zero. To me the natural approach would be to use the divergenceless of $\mathbf{B}$.

Answer 2

For static fields this can be done using Amperes circuit law.

$$ \nabla\times\vec{H}=\nabla\times(H_\phi(r)\hat{\phi}+H_z(r)\hat{z})=-\frac{\partial H_z}{\partial r}\hat{\phi}+r^{-1}\frac{\partial (r H_\phi)}{\partial r}\hat{z}\overset{!}{=}\vec{j}=j_0\hat{z} $$ And therefore by orthogonality $$ \frac{\partial H_z}{\partial r}=0 \rightarrow H_z(r)=C \quad \forall r $$ Enforcing a vanishing magnetic field infinitely far away from the current, a condition which every physically sensible solution should satisfy, one finds $$H_z(r)=0 \quad \forall r$$

It shall be stated, that this analysis can be extended for a time varying current (and therefore fields) in the region of the current carrying media (in the cable), assuming ohms law applies there and the conductivtiy is non-vanishing. For this convert the circuit law to frequency space and replace $\vec{E}(\vec{r},\omega)=\frac{\vec{j}(\vec{r}, \omega)}{\sigma(\omega)}$. This does not work outside the cable where $\sigma=0$.