Final answer:
The magnetic force is directly proportional to particle velocity in a magnetic field. The magnitude of force is the same for electrons and protons moving at the same velocity but applied in opposite directions. The magnitude and direction of a magnetic field affect the force on a charged particle.
Step-by-step explanation:
The relationship between the magnetic force and particle velocity is that they are directly proportional. The magnetic force on a charged particle, like an electron or proton, moving in a magnetic field is given by the equation F = qvBsinθ, where q is the charge of the particle, v is the velocity of the particle, B is the magnetic field strength, and θ is the angle between the velocity and the magnetic field. Therefore, if a particle's velocity increases while moving in a constant magnetic field, the magnetic force exerted on the particle also increases proportionally, assuming the angle remains constant.
When comparing an electron and a proton moving with the same velocity in a constant magnetic field, the magnitude of the force will be the same, but it will be applied in opposite directions due to their opposite charges. Their accelerations will differ because of their different masses. Increasing the magnetic field's magnitude does increase the magnetic force if the velocity and angle remain constant. However, changing the direction of the magnetic field changes the angle θ, which can change the force if it leads to a change in the sine of the angle.