Resultant Vector of magnetic fields

[DaGza]

For what reason is a magnetic force vector perpendicular to the magnetic field vector and the velocity vector of a charged particle? I know F=qvBsin and how to compute cross products but why does the force make a particle move in that particular direction? Is relativity involved?

[HallsofIvy]

No, relativity is not involved.

The only reasonable way to answer "For what reason is a magnetic force vector perpendicular to the magnetic field vector" is "that's the way it is defined". A magnetic "field" is not a "physical" thing- it is simply a way of caculating the force on an object at any point. Experimentally, it can be seen that the force on a charged particle in a magnetic field depends upon its velocity: the velocity vector has to be taken into account. About the only simple way of "multiplying" two vectors to get a third vector is cross product: the magnetic field is DEFINED in such a way that the cross product gives the correct direction.


(Well, in a sense relativity is involved- because the magnetic force depends on velocity, it would seem that experiments with magnetic fields should violate "Gallilean Relativity"- that no experiment in a closed room will distinguish between not moving or moving at a constant velocity. That lead to the Michelson-Morley experiment to try to do such an experiment which in turn lead to "Einsteinian Relativity".)

[pmb_phy]

No, relativity is not involved."

That's debatable. Many physicists hold that the Lorentz Force can be dervived from Coulomb's law using relativity. However I don't believe that's true myself. To do that one starts with Maxwell's equations and relativity. One then derives the transformation law for both the electric and magnetic fields and the force. It is then shown that the Lorentz force results. However MAxwell's equations are based on laws in which dictate that the magnetic force is qvxB.

It should be noted that this type of phenomena is not limited to EM. In fact the Coriolis force is of this type in that the Coriolis force proportional to the cross product of velocity and the angular velocity of the frame of reference.

In fact for any system for which there is a Lagrangian L = T - U where U is the generalized potential where U is linearly proportional to velocity. The force of such a system is of the same form as the Lorentz force.