- #1

- 19

- 0

## Main Question or Discussion Point

Hi all, I ran into a bit of an issue trying to figure out how to properly differentiate the magnetic force due to particle interactions. To be specific, I'm actually looking for the time derivative of acceleration (jerk) due to the magnetic force, but it's essentially the same problem.

For the record, this is not homework at all. It's something I require for a n-body simulation I'm doing, and I need to calculate jerk to do higher order integrations.

I know that the magnetic force is, in expanded form:

[tex]\vec{F} = \frac{\mu_{0} q_{1} q_{2}}{4\pi r^{2}} \vec{v_{1}} \times (\vec{v_{2}} \times \hat{r}) = \frac{\mu_{0} q_{1} q_{2}}{4\pi r^{2}}((\vec{v_{1}} \cdot \hat{r}) \vec{v_{2}} - (\vec{v_{1}} \cdot \vec{v_{2}}) \hat{r})[/tex]

Where r-hat is the unit vector pointing from particle 2 to particle 1, and r is the distance between the two particles.

I don't even know where to begin to do this. I understand I need the chain rule but when I tried doing the same for a simpler vector (gravitational acceleration), the actual result ended up having a bunch of dot products in there that I had no idea how to get.

Can anyone guide me in the right way?

For the record, this is not homework at all. It's something I require for a n-body simulation I'm doing, and I need to calculate jerk to do higher order integrations.

I know that the magnetic force is, in expanded form:

[tex]\vec{F} = \frac{\mu_{0} q_{1} q_{2}}{4\pi r^{2}} \vec{v_{1}} \times (\vec{v_{2}} \times \hat{r}) = \frac{\mu_{0} q_{1} q_{2}}{4\pi r^{2}}((\vec{v_{1}} \cdot \hat{r}) \vec{v_{2}} - (\vec{v_{1}} \cdot \vec{v_{2}}) \hat{r})[/tex]

Where r-hat is the unit vector pointing from particle 2 to particle 1, and r is the distance between the two particles.

I don't even know where to begin to do this. I understand I need the chain rule but when I tried doing the same for a simpler vector (gravitational acceleration), the actual result ended up having a bunch of dot products in there that I had no idea how to get.

Can anyone guide me in the right way?