# Why would charged particles travel to the poles of a planet given the magnetic field is like a bar magnet's?

I've been told that the magnetic field of a planet (say, Earth) causes charged particles traveling towards it to go toward the poles. Why? $$\mathbf{F_m} =q(\mathbf{v} \times \mathbf{B})$$ The above would say that given a magnetic field that points towards the poles, the charged particles experience a force in a direction perpendicular to the direction of the poles. And I also just can't imagine when the magnetic field lines we're usually shown would lead a particle to the poles. Would anyone mind explaining this?

The magnetic field for a dipole moment $\vec m$ $${\vec B}(\vec m,\vec r)~=~\frac{\mu_0}{4\pi}\frac{3(\hat m\cdot\hat r)\hat r~-~\vec m}{r^3}.$$ The magnetic moment we orient along the $z$ axis and $\vec r$ is the radial vector to a point in space. For a charged mass at that point in space moving with velocity $\vec v$ we have of course the Lorentz force $\vec F~=~q\vec v\times\vec B$. The motion is going to be a bit complicated, but we can look at the relevant part of the magnetic field which is $3(\hat m\cdot\hat r)\hat m~-~\vec m$ Let us assume the particle has a trajectory perpendicular to the magnetic moment. We then clearly have a component $\vec m\times\vec v$ to the force. The magnitude of this force is then going to be determined by the $\vec m\cdot\hat r$. So if the particle is on the equatorial plane of the dipole the force is smaller than if it is coming in at a plane tangent to the poles. Also note there is a difference in sign.