Motion of Particle in a Magnetic Field

[SoberSteve2121]

F=q(v x B)
If the velocity is in the i direction and the magnetic field is in the -k direction prove using Newtons Second Law that the trajectory of this charge must be circular?

[James R]

Consider the cross product first:

i \times -k = j

Therefore, the force is in the j direction, perpendicular to the velocity. By Newton's 2nd law, the acceleration is in the same direction, perpendicular to the velocity at all times. Therefore, the force cannot change the magnitude of the velocity - only its direction. Since the acceleration has constant magnitude, the charge moves in a circle at constant speed.