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.

 

[R0man]

Using Newton's Second Law, we know that the net force on a particle is equal to the mass of the particle times its acceleration. In this case, the net force is the magnetic force (F=q(v x B)) and the acceleration is equal to the change in velocity over time.

If we break down the equation for the magnetic force, we can see that the magnitude of the force is equal to the charge of the particle (q), the magnitude of the velocity (v), and the magnitude of the magnetic field (B). The direction of the force is perpendicular to both the velocity and the magnetic field, as shown by the cross product (v x B).

In this scenario, we have a velocity in the i direction and a magnetic field in the -k direction. This means that the force will be in the j direction, which is perpendicular to both the velocity and the magnetic field.

Now, if we consider the acceleration of the particle, we can see that it will always be directed towards the center of the circle. This is because the velocity is always changing in direction due to the force being perpendicular to it. By Newton's Second Law, the acceleration is directly proportional to the force, so it will also be directed towards the center of the circle.

Therefore, we can conclude that the trajectory of this charge must be circular, as the force and acceleration are always directed towards the center of the circle. This is known as uniform circular motion, where the velocity is constantly changing in direction but the speed remains constant.