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

  1. Nov 27, 2013 #1
    1. The problem statement, all variables and given/known data

    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)

    2. Relevant equations

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

    3. 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!
     
  2. jcsd
  3. Nov 27, 2013 #2

    pasmith

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    Homework Helper

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

    [tex]\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}}[/tex]

    To determine [itex]\nabla \cdot \mathbf{F}[/itex] you must first determine the components of [itex]\mathbf{F}[/itex]. The unit vectors are orthogonal, so [itex]F_r = \mathbf{F} \cdot \hat{\mathbf{r}}[/itex] etc. We see then that [itex]F_\phi = F_z = 0[/itex] and
    [tex]
    F_r = \frac {a \cos \phi}r.
    [/tex]
     
  4. Nov 27, 2013 #3
    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?
     
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