# Biot-Savart Law: infinity wire

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1. May 20, 2016

### Aroldo

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 $$I$$

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

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:
$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_{-\infty}^\infty{\frac{d\mathbf{l'} \times (\mathbf{r} - \mathbf{r'})}{|(\mathbf{r} - \mathbf{r'})|^{3/2}}}$$


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

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \frac{2 I}{s} \hat{\phi}$$

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?

2. May 20, 2016

### TSny

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

3. May 20, 2016

### Aroldo

Thank you a lot!