Question

really easy question.....
if velocity of a charged particle is in x and y components, and magnetic field, is in z direction, then the charged particle won't follow a helical path, right? just wanted to be sure.
thanks! :)

Answer 1

Force $\vec{F}$ on a charged particle which is moving with velocity $\vec{v}$ in a magnetic field  $\vec{B}$ is given by :
       
  $\vec{F}=q\ \vec{v}\ X\ \vec{B}=q\ (v_x\ \hat{i} + v_y\ \hat{j}) X B_z \hat{k} \\\\=q\ B_z\ (v_y \hat{i} - v_x\ \hat{j})$

Since, the velocity and magnetic fields are perpendicular, the particle experiences a constant force, perpendicular to B and v.  There is a constant acceleration in x-y plane always perpendicular to the velocity.  Hence the charged particle moves in a circular path of a fixed radius and center.

If the charged particle has a velocity component in z direction also, then it will move in a helical path.