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Biot-Savart Law: infinity wire

  1. May 20, 2016 #1
    Hey!


    1. The problem statement, all variables and given/known data

    One must simply calculate the magnetic field at a distance s to the wire, which carries a steady current [tex]I[/tex]

    2. Relevant equations
    Should I write the point vector as:
    [tex]\mathbf{r} = s\hat{s} + \phi \hat{\phi} + z \hat{z}[/tex]
    or
    [tex]\mathbf{r} = s\hat{s} + z \hat{z}[/tex] ?

    3. The attempt at a solution
    I am not solving it as the author does. I am trying to use spherical coordinates, therefore I am writing:
    [tex]\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_{-\infty}^\infty{\frac{d\mathbf{l'} \times (\mathbf{r} - \mathbf{r'})}{|(\mathbf{r} - \mathbf{r'})|^{3/2}}}[/tex]
    [tex] [/tex]

    Where:
    [tex]d\mathbf{l'} = dz \hat{z} [/tex]
    [tex]\mathbf{r} - \mathbf{r'} = s \hat{s} + z \hat{z} - z'\hat{z} = s \hat{s} + (z-z')\hat{z}[/tex]

    and the answer is fine:
    [tex]\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \frac{2 I}{s} \hat{\phi}[/tex]

    But, if I consider the vector as:
    [tex]\mathbf{r} = s\hat{s} + \phi \hat{\phi} + z \hat{z}[/tex]

    (it seems to me more general) The answer has a component in the s-direction, which is incorrect.

    Please, what is wrong in my reasoning?
     
  2. jcsd
  3. May 20, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    If you draw a figure that attempts to show how the three terms ## s\hat{s} + \phi \hat{\phi} + z \hat{z}## combine to give ##\mathbf{r}##, you'll see why the ## \phi \hat{\phi}## term should not be included. It is important to understand the meaning of the unit vector ##\hat{s}##.

    Note that in spherical coordinates ##(r, \theta, \phi)##, the position vector is just ##\mathbf{r} = r \hat{r}##. It is not ##\mathbf{r} = r \hat{r} + \theta \hat{\theta}+\phi \hat{\phi}##
     
  4. May 20, 2016 #3
    Thank you a lot!
     
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