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The following image is from Halliday and Resnick, Fundamentals of Physics. The magnetic field at P due to the current element is shown as being directed into the page. The textbook and other sources state that the direction of the cross product in the Biot-Savart law gives the direction of the magnetic field at that point, but I can't see how this can be deduced from the formula alone. For the electric field created at a radial distance by a point charge, there is a radial unit vector which tells us the direction of the field along that line. I don't see which part of the Biot-Savart law contains information about the direction of the magnetic field at a radial distance away from the current element. Is this a generally accepted property of cross products, or is there no way to express this information in a formula?

enter image description here

Thomas Fritsch
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Piksiki
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  • What does "generally accepted" mean?? The cross product takes two vectors and gives another. Vectors have a magnitude and direction. If the cross product didn't produce a direction then it wouldn't be the cross product; it's not optional. (Setting aside the dubious physical provenance of the cross product.) – HTNW Jul 30 '22 at 14:54
  • @HTNW, my question is not about the cross product. I am wondering why the result of the cross product holds for the direction of the field at a radial distance from the point for which this value is calculated, i.e. where does the formula state that the cross product direction holds for the direction of the field along the radial vector. – Piksiki Jul 30 '22 at 15:31
  • Related : Magnetic field due to a single moving charge. – Frobenius Jul 31 '22 at 05:55

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According to the Biot-Savart law the current element $I\ d\vec{s}$ gives this contribution to the magnetic field $\vec{B}$ at point $P$: $$d\vec{B}=\frac{\mu_0}{4\pi}\frac{I\ d\vec{s}\times\vec{r}}{r^3}$$

From this formula you see that $d\vec{B}$ has the direction of $d\vec{s}\times\vec{r}$. Given the direction of $d\vec{s}$ (the green arrow) and $\vec{r}$ (the red arrow) and using the right-hand-rule for the direction of the cross product you find $d\vec{s}\times\vec{r}$ and thus $d\vec{B}$ is pointing into the page.

Thomas Fritsch
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  • I understand what the cross product is used for, but where does the formula contain the information that the result of the cross product gives the field's direction along any point on the radial vector? Once the cross product is calculated, there is no indication of the direction in which this result applies for the magnetic field direction by looking at the formula alone. – Piksiki Jul 30 '22 at 15:40
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    @Piksiki Uh, by saying that the magnetic field is ($=$) the cross product of current and displacement, up to scaling? – HTNW Jul 30 '22 at 15:43
  • @Piksiki It is a vector equation. The vector on the RHS is equal to the vector on the LHS (including the direction). – Thomas Fritsch Jul 30 '22 at 15:48
  • @HTNW Do you mean this in the sense that it is determined experimentally, or is there a nuance I am not aware of? – Piksiki Jul 30 '22 at 16:39
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I'm not sure I understand the question... It's just a basic property of the cross-product: $\vec{a}\times\vec{b}$ is perpendicular both to $\vec{a}$ and $\vec{b}$ (except of course if $\vec{a}$ and $\vec{b}$ are colinear, in which case the product is simply zero).

If your question is more about why the magnetic field has this property, it's a bit more subtle. $\vec{B}$ is a pseudovector.

Miyase
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