Question

Consider a particle with unit charge q, and mass m, in a constant magnetic field B directed along the positive z–axis. The particle is subject to Lorentz force: F = q [v × B] , (1) where v is the velocity of the particle. Assuming that initially the particle was at the origin r(t = 0) = 0 with velocity v(t = 0) = (u, 0, w) determine the trajectory of the particle.

Answer 1

Answer:

it must be helical motion in which the charge particle will move uniformly along z axis and simultaneously it will move in circular path in xy plane.

Explanation:

Magnetic field is along z axis while velocity is in x-z plane

so we will have

$F = q(\vec v \times \vec B)$

so here we can say

$F = q(u\hat i + w\hat k) \times (B \hat k)$

so we will have

$F = quB(-\hat j)$

so here the net force on the charge is perpendicular to its x directional velocity along - Y direction

So due to this component of motion it will move along a circle while other component of the motion will remain uniform always

So here it is combination of two parts

1) Uniform circular motion

2) Uniform motion

So we can say that it must be helical motion in which the charge particle will move uniformly along z axis and simultaneously it will move in circular path in xy plane.