What's Wrong with My Biot-Savart Law Calculation?

[Aroldo]

Hey!


1. Homework Statement
One must simply calculate the magnetic field at a distance s to the wire, which carries a steady current I

Homework Equations

Should I write the point vector as:
\mathbf{r} = s\hat{s} + \phi \hat{\phi} + z \hat{z}
or
\mathbf{r} = s\hat{s} + z \hat{z} ?

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:
\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}}}


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

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

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

(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?

 

[TSny]

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

(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 $\mathbf{r} = r \hat{r} + \theta \hat{\theta}+\phi \hat{\phi}$

 

[Aroldo]

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!