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There seems to be a pretty standard formula, that if a wire of length $\ell$ carrying current $I$ is immersed in a magnetic field $B$, then the magnitude of the magnetic force is $$F_B = I \ell B sin \theta$$, where the direction of $F_B$ is the direction of $\ell \times B$ (determined using right-hand rule).

But, these properties can all be summarized into the equation $$dB = \frac{\mu_0}{4\pi} \frac{Id \ell \times r}{r^2}$$

I understand the first equation, but don't get the second one. How would I even read this equation or apply it? Do I have to integrate first? How would this work?

Thank you in advance for any help/clarifications.

mathz2003
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    The second equation is the Biot-Savart law, which is a different concept from the formula for magnetic force. The Biot-Savart law gives magnetic field at any point due to current in wire; it does not give magnetic force acting on that wire. – Ján Lalinský Apr 21 '22 at 21:07
  • In addition to what Jan has stated, check this as well (also your expression for $dB$ needs to have an $r^3$ in the denominator if $r$ in the numerator is not a unit vector - remember that $B$ has an inverse-square behavior like the electric and gravitational fields). – joseph h Apr 21 '22 at 21:10
  • This is very interesting. My textbook put them both side-by-side and said one derives the other. Is there possibly some error and my textbook put the wrong formula? – mathz2003 Apr 21 '22 at 21:16
  • The point we are making is that one equation describes forces due to current carrying wires, and the second describes magnetic fields at points from the wire. Whether you can derive one from the other is another question. What textbook are you using? – joseph h Apr 21 '22 at 21:41
  • Related : Magnetic field due to a single moving charge. – Frobenius Apr 22 '22 at 09:19

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As the comments have already pointed out the two equations are related but not the same. The first equation represents the "magnetic force on a straight wire segment" due to a magnetic field $B$. That is to say, if you had a wire carrying current $I$ and "immersed it", as you put it, into a magnetic field of strength $B$, the force experienced by the wire is

$\vec{F} = I \vec{l} \times \vec{B} $

and the magnitude is given by $I l B sin \theta$, and the $sin \theta$ just comes from the cross product.

As you know $I$ is scalar but this force is a cross product so you use the direction the current is flowing in, denoted by $\vec{l}$, to figure out the direction. If needed I can provide the derivation of this equation, but I'd just be copy/pasting out of my book University Physics by Young and Freedman, Chp 27 section 27.6.

Now, if you are given a nice constant magnetic field, you can insert it into this equation and be done. But if you need to derive what the magnetic field is, due to some current carrying wire(s), solenoid, etc. you need to use Biot-Savart or Ampere's Law.

In general you get this from computing:

$\vec{B} = \frac{\mu_0}{4 \pi} \int \frac{I d\vec{l} \times \vec{r}}{r^2}$

So to get the force on a current carrying wire, from say another current carrying wire, you'd really need to do this:

$\vec{F} = I \vec{l} \times \frac{\mu_0}{4 \pi} \int \frac{I d\vec{l} \times \vec{r}}{r^2}$

tau1777
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