Biot-Savart Law: infinity wire

  • #1
14
0
Hey!


1. Homework Statement

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

Homework 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] ?

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?
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
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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.
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}##
 
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  • #3
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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}##
Thank you a lot!
 

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