# Curl of magnetic field (Ampère's law)

I'm a little bit confused about Ampère's law concerning the curl of a magnetic field: $$\vec{\nabla}\times\vec{B}(\vec{r}) = \mu_{0}\vec{J}(\vec{r})$$

with $$\vec{J}(\vec{r})$$ the volume current density in $$\vec{r}$$. How can this be true in the case of an infinite straight wire carrying a steady current $$\vec{I}$$ ? Clearly the magnetic field has nonzero curl everywhere but the volume current density is zero everywhere because there is no charge density anywhere except maybe along the line. Am I misunderstanding the notion of volume current density ?

• You can’t have a current without having a current density. You can have a current density without having a charge density. For example, if you have positive charge density (say, protons) sitting still, and an equal negative charge density of electrons, with some of them movng, then you have no net charge density but you do have a current density. The wire $does$ have a current density, Jun 22 '19 at 20:44
• Basically the density thing changes depending on the kind of thing it is. Here it should be current per area density thing. And we say that the wire has some cross ectional area Apr 20 '20 at 2:47

You can’t have a current without having a current density. But you can have a current density without having a charge density.

For example, if you have positive charge density (say, protons) sitting still, and an equal negative charge density of electrons, with some of them moving, then you have no net charge density but you do have a current density.

So... the wire does have a current density.

• No, you don’t understand curl. It doesn’t mean “curving field lines”. Just because field lines form circles around a wire, it doesn’t mean that the mathematical curl outside is nonzero. Try computing the field outside the wire, and compute its curl. Jun 22 '19 at 21:30
• Look up the expression for the curl in cylindrical coordinates $(\rho, \phi, z)$. Then compute it for $\vec{B}\sim\hat{\phi}/\rho$. It will be zero. Jun 22 '19 at 21:41
• It’s kind of similar to divergence. When you have a point charge, the electric field is radially outward. The field lines “diverge” from each other everywhere. But the mathematical divergence of the field is nonzero only at the charge. Jun 22 '19 at 21:43
• @einsteinwasmyfather I struggled with a similar concept as you are right now, but with divergence. The divergence of the electric field around a point particle is zero everywhere except at the position of the particle. Likewise, in this case, the curl of the magnetic field is exactly zero everywhere except on the wire. Jun 22 '19 at 21:52
• The curl is a measure of local rotation. As a way of observing its effects, one might place a small pinwheel at the location of $\vec{r}$ and observe it's motion from the force exerted by the field. If it begins to rotate, there is a non-zero curl at that point. The magnetic field everywhere not on the wire is locally a straight line. It is only exactly at the wire that a very small pinwheel will locally rotate from the magnetic field. Jun 22 '19 at 21:57

The actual curl is

$$\nabla \times\vec{B}(\vec{r}) = \mu_0\vec{J}(\vec{r})+\mu_0\epsilon_0\frac{\partial \vec{E}(\vec{r})}{\partial t}$$

However, commenters above have answered the question as to why your example is incorrect. There is a current density on the straight wire. You cannot have a current without a current density.

First, you are asking about Ampere's law in magnetostatics, and is definitely a valid question.

Second, you're right. Both sides of this equation are volume-based quantities. But when converting to integral form by using Stokes' theorem, you have to pick a contour and the surface enclosed by the contour. If you pick the contour as a circle rotating around the wire, then you arrive at the integral form the Ampere's law. The volume current density though may point in any direction, but only the portion that goes through the surface ends up contribute.

You are right, the curl of the magnetic field seems like zero everywhere for a wire current (when you directly calculate the differential value), but it is not zero at the wire location. You can apply Stokes theorem in cylindrical coordinate for a small circle around the wire to find it out. It is equal to I/S, where S is the area. When you make S infinitely small, it is approaching infinity. Therefore, the curl for the H field of an infinite wire at the wire location is a delta function along the wire. It is zero everywhere, except for when collocated with the wire.