Quick questions on Charged particles in magnetic fields - TCE Physics (Tasmania)

The force on a charge $q$ moving at speed $v$ through a field $B$ is $F = qvB\sin\theta$, maximum when the velocity is perpendicular to the field. This force is always perpendicular to the velocity, so it never speeds the charge up or slows it down; it only changes the direction. A force of constant size always perpendicular to the motion is exactly what produces uniform circular motion.

Substituting $v = \dfrac{2\pi r}{T}$ into the radius equation and simplifying gives the period:

When a charge enters at an angle to the field, only the perpendicular component $v_\perp = v\sin\theta$ drives the circular motion, while the parallel component $v_\parallel = v\cos\theta$ carries the charge steadily along a field line. The result is a helix (a spiral) whose radius is set by $v_\perp$ and whose pitch (the distance advanced per turn) is set by $v_\parallel$. This is exactly how charged particles spiral along Earth's magnetic field lines in the Van Allen belts and funnel toward the poles to create aurorae.