Final answer:
The force acting on a charged particle in a magnetic field is F = qvB sin θ. For circular motion, the charge needs to move perpendicularly to a uniform magnetic field so that F becomes the centripetal force (F = mv²/r), resulting in uniform circular motion with constant speed.
Step-by-step explanation:
Force Acting on a Charge in a Magnetic Field
The expression for the force F acting on a particle with charge q and mass m moving with velocity V in a magnetic field B is described by the equation F = qvB sin θ, where θ is the angle between the velocity of the charge and the magnetic field. For the charged particle to move in a circular path, this force must act as the centripetal force, which means the charged particle’s velocity must be perpendicular to the magnetic field, making sin θ equal to 1. Hence, the force required for circular motion is F = qvB.
The direction of the magnetic force can be determined using the right-hand rule. Curl the fingers of your right hand from the direction of velocity to the direction of the magnetic field (V to B), and your thumb will point in the direction of the magnetic force (F) acting on the particle.
For circular motion to occur, the particle must be moving in a plane perpendicular to the uniform magnetic field. If this is the case, the magnetic force provides the centripetal force necessary for circular motion, according to F = mv²/r, where r is the radius of the circular path. With the condition of perpendicular motion, velocity V maintains constant magnitude because the magnetic force is perpendicular to it and does no work, thus not changing the kinetic energy or the speed of the particle.