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Finding the magnetic field of an infinite cylindrical wire.

  1. Nov 28, 2014 #1
    1. The problem statement, all variables and given/known data
    An infinite cylindrical wire of radius ##R## carries a current per unit area ##\vec{J}## which varies with the distance from the axis as ##J(s)=ks^2\hat{z}## for ##0<s<R## and zero otherwise where k is a constant.
    Find the magnetic field ##\vec{B(s)}## in all space.
    2. Relevant equations
    ##\oint B \cdot dl=\mu_0I_{enc}##
    ##\nabla \times B=\mu_0 \vec{J}##
    3. The attempt at a solution

    ##\oint B \cdot dl=B2\pi s##, ##I_{enc}=
    \Bigg\{
    \begin{array}{lr}
    \frac{\pi ks^4\hat{z}}{2}, 0<s<R\\
    \frac{\pi k R^4\hat{z}}{2},s \ge R
    \end{array}##
    So,
    ##\vec{B}=
    \Bigg\{
    \begin{array}{lr}
    \frac{ks^3\hat{z}]\mu_0}{4}, 0<s<R\\
    \frac{kR^4\hat{z}\mu_0}{4s}, s \ge R
    \end{array}##.
    However, when I calculate the curl of ##\vec{B}##, I don't get back ##\vec{J}##. Where am I wrong?
     
  2. jcsd
  3. Nov 28, 2014 #2

    vela

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    For one thing, ##\vec{B}## doesn't point in the ##\hat{z}## direction.
     
  4. Nov 28, 2014 #3
    It's radially isn't it?
     
  5. Nov 28, 2014 #4

    vela

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    No. You must have seen figures depicting the magnetic field around a wire in your book, right? Use the right-hand rule to determine the direction.
     
  6. Nov 28, 2014 #5
    Okay. So if I use ##\vec{J} \times \vec{r}## to find the direction of the magnetic field inside the wire, then the magnetic field should be in the ##\hat{\phi}## direction?
     
  7. Nov 29, 2014 #6

    vela

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    Yup.
     
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