What if curl B = 0 AND div B = 0

[Henry Morton]

Well, the reason I'm asking this is because we recently did a problem in my class where we were supposed to show some vector identity, with the conditions that both

curl B = 0

and

div B = 0

The problem was really about the maths, but it was phrased as if the field were a magnetic field. Now, I'm just wondering what the physical implications of this would be. Obviously if B is a magnetic field then div B = 0 is just one of Maxwells equations, but what if curl B = 0 as well? If B=0 both conditions are satisfied, but is there a non-zero vector/magnetic field that satisfies these conditions?

I'm really at a loss here...

 

[Jano L.]

Hi Henry,
welcome to PF. The equation curl B = 0 holds only if the field is constant in time and in places where the current density is zero. For example, the magnetic field has zero curl when it is due to non-moving permanent magnet, or due to constant current in wires.

Jano

 

[maCrobo]

Hi!

Think about it mathematically to be sure of your result, so rewrite curl and div as $\nabla$. Then $\nabla \times B=0$ and $\nabla \cdot B=0$.
This means that B must be parallel and perpendicular to the same vector, then the unique solution is B=0.

In terms of Physics you that divB=0 by MAxwell, than if curl B=0 you can be sure about the fact that there's no magnetic field.

 

[chrisbaird]

Hi!

Think about it mathematically to be sure of your result, so rewrite curl and div as $\nabla$. Then $\nabla \times B=0$ and $\nabla \cdot B=0$.
This means that B must be parallel and perpendicular to the same vector, then the unique solution is B=0.

In terms of Physics you that divB=0 by MAxwell, than if curl B=0 you can be sure about the fact that there's no magnetic field.

Actually...
If $\nabla \times \textbf{B}=0$, then from the vector identity: $\nabla \times \nabla ψ=0$, we can write $\textbf{B} = - \nabla ψ$.

If we also have $\nabla \cdot \textbf{B}=0$, then substituting in, we find:
$\nabla^2 ψ=0$

This is the Laplace equation, and it has many possible solutions depending on the boundary conditions.

Perhaps thinking in terms of magnetic field line diagrams is more helpful. A diverging field means the field lines emanate out in all directions from a source point. A curling field means each field line forms a closed loop by connecting back to itself. So a non-diverging, non-curling field just means that the field lines don't emanate from a point or connect back to themselves. There are many ways to draw non-zero field lines that don't curl or diverge.

In general, every vector field has potentially three natures that are, in a sense, "orthogonal": (1) it's curling nature, (2) it's diverging nature, and (3) it's relaxation nature. I call it relaxation, because the solutions to the Laplace equation look like sheets of rubber stretched across a boundary that have relaxed to their most natural shape.

 

[Henry Morton]

Hi Henry,
welcome to PF. The equation curl B = 0 holds only if the field is constant in time and in places where the current density is zero. For example, the magnetic field has zero curl when it is due to non-moving permanent magnet, or due to constant current in wires.

Jano

Thanks. So if curl B = 0 at some point basically means that there is no source of magnetic field there? And then if curl B = 0 everywhere, it means there is no magnetic field? I guess this is what maCrabo means.

I think a problem for me is that I have always been used say that if curl B = 0 that means that B is a conservative, but it's nonsense to talk of a conservative magnetic field since it cannot do work in the first place right?

 

[chrisbaird]

If the magnetic field had no curl and no divergence across the entire universe, then from a physical perspective B = 0.

 

[maCrobo]

Chrisbaird you are absolutely right, but in that "relaxation" state vectors that makes the field are kind of disordered in the space so not to have any sort of behavior like curling or diverging, then we can say the overall result is approximately zero in our set (portion of space).

Practically a sort of relaxation arrangement can be found in a system full of atoms spread in such a way that their magnetic moments adds to zero (or almost zero :P ). Anyway from the macroscopic point of view the result can be seen as zero.

 

[Matterwave]

If the magnetic field had no curl and no divergence across the entire universe, then from a physical perspective B = 0.

A constant B everywhere in space also solves the math equation, but I suppose it would be physically hard to justify.

 

[chrisbaird]

Chrisbaird you are absolutely right, but in that "relaxation" state vectors that makes the field are kind of disordered in the space so not to have any sort of behavior like curling or diverging, then we can say the overall result is approximately zero in our set (portion of space).

Practically a sort of relaxation arrangement can be found in a system full of atoms spread in such a way that their magnetic moments adds to zero (or almost zero :P ). Anyway from the macroscopic point of view the result can be seen as zero.

No. "Relaxed" vector fields (non-diverging, non-curling) are not disordered. Quite the opposite - they are the most ordered fields of all, and can definitely be non-zero. Consider the two magnets in the diagram below. In the volume of interest alone, shown as the blue shaded box, the magnetic field lines are neither diverging nor curling, and yet they are very ordered and very non-zero.