Magnetic force between two charged particles? I know the gravitational force between two particles with masses $m_1$, $m_2$:
$$
\vec{F} = \frac{Gm_1m_2}{|\vec{r}|^2} \hat{r}
$$
And I know the electrical force between two particles with charges $q_1$, $q_2$:
$$
\vec{F} = -\frac{Kq_1q_2}{|\vec{r}|^2} \hat{r}
$$
($\vec{r}$ is the position vector of particle 2 from the referential of particle 1 and $\hat{r} = \frac{\vec{r}}{|\vec{r}|}$)
I've been looking for an expression like these for the magnetic force since 2012... Then I found Physics Stack Exchange. Does anybody know if such formula exists?
 A: What you want is essentially the Biot-Savart Law.
For a point charge that is moving slowly compared to the speed of light (which is also a condition for the Couloumb law that you give to be true, by the way), Biot-Savart says that a point charge makes a magnetic field like:
$\vec{B}=\frac{\mu_0}{4\pi}q_{1}\vec{v_1}\times\frac{\hat{r}}{r^2}$,
where $\vec{v_1}$ is the velocity of particle 1 and $q_{1}$ is its charge.
Then, the force particle two feels from it is the Lorentz force,
$\vec{F_2}=q_{2}\vec{v_2}\times\vec{B}$,
where $\vec{v_2}$ is its own velocity and $q_{2}$ its charge.
Put them together and you get the magnetic force one particle feels from the other,
$\vec{F_{1 \rightarrow 2}}=\frac{\mu_0 q_{1}q_{2}}{4\pi r^2}\vec{v_2}\times\{\vec{v_1}\times\hat{r}\}$
So it is a force that is very direction-dependent, unlike the other two formula you give: it depends on the velocities of each particle, both directions and magnitudes, as well as how these directions compare to the direction of the line that separates the two particles. For a given combination of these directions and speeds, it falls off as r^2 just like the other two forces.
A: Probably you are interested in the magnetic force between two moving charges which is
$$\vec{F}=\frac{\mu_0}{4\pi}\frac{q_1 q_2}{r^2}\vec{v}_1\times (\vec{v}_2\times\hat{r})$$
A: If there where magnetic monopoles, the force between them in static conditions would be exactly the same as that described by Coulomb. You'd need to replace the electric charges with the magnetic charges, and possibly the universal constant as well. All of this is just a consequence of the symmetry of Maxwell's equations with a magnetic source.
A: To clarify upon the other answers: There is no magnetic force between non-moving charged particles. Other answers posted here have shown that if there is motion between two charged particles, there will be a magnetic force between them given by the Biot-Savart law.
Actually, the Biot-Savart law covers both moving and non-moving cases. If the velocities are equal (i.e. same speed, same direction), the force will be zero. Two non-moving particles are considered to have equal velocity in some frame of reference.
Additionally, our understanding of electrodynamics tells us that two non-moving magnetic monopoles should experience a force analogous to the electric force between two electric charges. These magnetic monopoles would possess some kind of "magnetic charge" analogous to electric charge. However, magnetic monopoles are not observed in nature; only dipoles are found.
Some of the comments have correctly stated what we observe empirically: There is an electric force between charges, and a magnetic force between moving charges.
