Testing to see if the vector field could be a magnetic field.

[Michael 37]

Homework Statement

By considering its divergence, test whether the following vector field could be a magnetic field:
F=(a/r) cos∅ r Where a is constant. NOTE( the 'r' has the hat symbol ontop if it, unit vector i think)

Homework Equations

You may use that is cylindrical co-ordinates:
∇.F=(1/r)∂/∂r(rFr) + (1/r)∂F∅/∂∅ + ∂Fz/∂z

The Attempt at a Solution

I tried to differentiate F with repect to r but the 'rFr' threw me! Then differentiating with respect to ∅ then z. But I still don't understand what they want me to test. I am lacking not only in the mathematical side to the question but also the theory. Please help, been at it for days!

 

[pasmith]

Homework Statement

By considering its divergence, test whether the following vector field could be a magnetic field:
F=(a/r) cos∅ r Where a is constant. NOTE( the 'r' has the hat symbol ontop if it, unit vector i think)

Homework Equations

You may use that is cylindrical co-ordinates:
∇.F=(1/r)∂/∂r(rFr) + (1/r)∂F∅/∂∅ + ∂Fz/∂z

The Attempt at a Solution

I tried to differentiate F with repect to r but the 'rFr' threw me! Then differentiating with respect to ∅ then z. But I still don't understand what they want me to test. I am lacking not only in the mathematical side to the question but also the theory. Please help, been at it for days!

Magnetic fields are divergence-free (recall the Maxwell equation $\nabla \cdot \mathbf{B} = 0$), so if $\nabla \cdot \mathbf{F} \neq 0$ then $\mathbf{F}$ cannot be a magnetic field. But if $\nabla \cdot \mathbf{F} = 0$ then it could.

$$\mathbf{F} \equiv F_r\,\hat{\mathbf{r}} + F_\phi\,\hat{\mathbf{\phi}} + F_z\,\hat{\mathbf{z}} = \frac ar \cos \phi\,\hat{\mathbf{r}}$$

To determine $\nabla \cdot \mathbf{F}$ you must first determine the components of $\mathbf{F}$. The unit vectors are orthogonal, so $F_r = \mathbf{F} \cdot \hat{\mathbf{r}}$ etc. We see then that $F_\phi = F_z = 0$ and
$$F_r = \frac {a \cos \phi}r.$$

 

[Michael 37]

Is the reason for Fϕ=Fz=0 because in the F function, there are no ∅ or z components?

So now all that is left is Fr = (a/r) cos ∅ so do i need the divergence of that (in other words do i differentiate that with respect to r?) And if it = 0, it may be a magnetic field and if it doesn't = 0 it can't be a magnetic field?