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I understand that when a charged particle enters a magnetic field with a velocity $v$, with a angle $\theta$ within the field lines, its perpendicular and parallel velocity components generate, respectively, a circular and transverse motion, causing the particle to have a helical motion. However, in this development, it is often assumed that the magnetic field is uniform, and, having this premise established, it is not difficult to mathematically demonstrate why the particle describes a helical motion, one very nice and simple way is the first answer on this post: Helical motion of charged particle in external magnetic field.

But let's take the magnetic mirror as an example. In it, the magnetic field is variable, and yet the particle describes a helical movement along $\hat{z}$!

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"Geometrically", it's not difficult to understand, the particle still have the components of the velocity producing the respective motions, after all. But I've not been able to rigorously demonstrate this fact, mathematically.

Summarizing, I want to demonstrate, mathematically, why particles, in non-uniform magnetic field regions, continue to describe helical movements, when entering the field with a angle $\theta$.

honeste_vivere
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Johann Wagner
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It doesn't do exactly helical motion, since the idea of the magnetic mirror is that the particle eventually turns around and bounces between the ends. The statement is that if the B field is approximately constant, then we will get approximately helical motion. If it is uniform clearly we get helical motion, then the only part left is to see what the small force is that makes it not exactly helical. This if from a force along mirror axis, or $F_{z}$. The Lorentz force is $$\vec F=q \vec v \times \vec B$$ Assume $v_z \ll v_\perp$, where $v_\perp$ is the gyroscopic motion $v_\perp=r \omega_c$, for $\omega_c=qB/m$. Then if the $B$ tilts radially inward (it has to since $\nabla \cdot B=0$), or $\vec B = -B_r \hat r + B_z \hat z$, we get a force backwards $$\vec F = -q v_\perp B_r \hat z.$$ So there is a force away from the ends, but if this is very small, then the motion is helical.

If it is of interest: from a many step derivation, I get the orbit area, A, changes as $$\dot A = -3 \frac{1}{B_z}\frac{d}{dz} B_z v_z A,$$ so roughly speaking this analysis is OK as long as the area changes slowly compared to the orbital period, or $$\omega_c \gg \frac{1}{B_z}\frac{d}{dz} B_z v_z. $$

Zach Johnson
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Summarizing, I want to demonstrate, mathematically, why particles, in non-uniform magnetic field regions, continue to describe helical movements, when entering the field with a angle $\theta$.

As I showed/stated in https://physics.stackexchange.com/a/670591/59023 and https://physics.stackexchange.com/a/671056/59023, the critical piece of information is how fast the field changes relative to one of the periodic motions of the particle. If the gradient scale length is short and the field gradient strong, the particles can deviate from the usual circular gyration as illustrated in the movies at https://svs.gsfc.nasa.gov/4513. These are charged particles incident on a collisionless, magnetized shock wave that are undergoing something called shock drift acceleration or SDA. Under the right conditions, the particle orbits can actually trace out a cycloid. The particles are undergoing a gradient drift (e.g., see https://physics.stackexchange.com/a/556682/59023) while also being accelerated by an electric field due to a Lorentz transformation.

Regardless, the short answer to your question is just the Lorentz force. In the absence of an electric field, the particle only experiences a force orthogonal to both the magnetic field, $\mathbf{B}$, and the particle velocity, $\mathbf{v}$.

Now suppose there is a static electric field (and static magnetic field) present but that $\lvert \mathbf{B} \rvert > \lvert \mathbf{E} \rvert$. Under this scenario, the particle would merely ExB-drift (e.g., see https://physics.stackexchange.com/a/448523/59023) in a direction orthogonal to both $\mathbf{E}$ and $\mathbf{B}$. Conversely, if $\lvert \mathbf{B} \rvert < \lvert \mathbf{E} \rvert$ then there will be a reference frame where the particle is acted upon by a purely electrostatic field and the particle trajectory would be a hyperbolic path.

honeste_vivere
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