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This question is a continuation of my previous post. Using this magnetic field equation in vector polar coordinates, I was curious to test Maxwell's equation $$\bigtriangledown \times \vec{B} = \frac{\partial E}{\partial t} + \vec{J} \, .$$

The polar coordinates of the magnetic field with respect to $|r|$ is

$$ \vec{B} = \left ( 2 |u| \frac{\mu \cos \theta}{4 \pi |\vec{R}|^3}\right )\hat{r} + \left (|u| \frac{\mu \sin \theta}{4 \pi |\vec{R}|^3}\right )\hat{\theta} + 0\hat{\phi} \, . $$

However, when I take the $\bigtriangledown \times B$ I get zero.

Here are my steps:

In polar coordinates

$$ \bigtriangledown \times \vec{B} = \left[ \frac{\partial}{\partial \theta}B_\phi \sin\theta - \frac{\partial}{\partial \phi}B_\theta \right]\hat{r} + \frac{1}{r}\left[ \frac{1}{\sin\theta}\frac{\partial}{\partial \phi}B_r - \frac{\partial}{\partial r}rB_\phi \right]\hat{\theta} + \frac{1}{r}\left[ \frac{\partial}{\partial r}rB_\theta - \frac{\partial}{\partial \theta}rB_r \right]\hat{\phi} \, . $$

Luckily the magnetic field vector, $\hat{\phi} =0$ which simplifies our equation to

$$ \bigtriangledown \times \vec{B} = \frac{1}{r}\left[ \frac{\partial}{\partial r}rB_\theta - \frac{\partial}{\partial \theta}rB_r \right]\hat{\phi} \, . $$

Solving for the partial derivatives inside the brackets leave me with

$$ \bigtriangledown \times \vec{B} = \frac{1}{|R|}\left[ -\frac{\mu_o|u|\sin\theta}{2 \pi|R|^3} +\frac{\mu_o|u|\sin\theta}{2 \pi|R|^3}\right]= 0 \, . $$

I can't even continue with this equation, because it's simply zero. This doesn't make sense to me. Can someone please explain how I obtained zero? I understand putting those vector components through $\bigtriangledown \cdot \vec{B}$ gives me 0, but not this.

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  • $\begingroup$ @garyp The vector component $\hat{\phi}$ is zero in a polar coordinates simply because the magnetic field does not change with respect to $\hat{\phi}$, equi-potential lines. Therefore anything in the original polar curl definition that contains $\hat{\phi}$ is 0, leaving you with the 'simplified'. $\endgroup$ – macas Apr 20 '17 at 20:47
  • $\begingroup$ Yeah, I figured that out and deleted my comment. Sorry about that. $\endgroup$ – garyp Apr 20 '17 at 20:48
  • $\begingroup$ No problem. Also to comment on your first comment. I care only about values $R>0$ $\endgroup$ – macas Apr 20 '17 at 20:49
  • $\begingroup$ In the region where you're calculating the field, the current is zero, so it makes sense that the curl should be zero. Similarly, the electric field of a point charge has zero divergence for $r \neq 0$, even though there is charge density somewhere. The magnetic field diverges at the point where the current is nonzero, so you have to make a more careful calculation. $\endgroup$ – Javier Apr 20 '17 at 21:44

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