A couple of questions about magnetic field and a charge moving in it?

Question

A charge is moving with a constant velocity and a magnetic field is perpendicular to this motion. we know there is a mag.force that is given as the cross product of B and v.
F=q(v x B)

OK, so my doubt is, if there is a force, then why isn't the particle accelerating?
if it is accelerating in some direction then who does the work in moving it?

Another thing is : if we equate centripetal force to the aforementioned force, we get:
qvB=m(v^2) / r

What will happen if we increase the magnetic field, will the radius decrease or velocity increase or both? Kindly reply.

Answer 1

Hello Hi My name is Daniela but you can call me dani, the force acting is named as Lorentz magnetic force and it is perpendicular to the direction of motion.

Hence work done = force * displacement * cos 90

So work done = 0

More over as the force is perpendicular to the direction of motion, no chance of accelerating the movement.

This perpendicular force is named as centripetal force and it would change the direction of motion and so the charged particle would go in a circular path.

Now as we increase the magnetic field, then the radius would decrease since no chance of changing the magnitude of velocity as there is no acceleration in the direction of velocity.

Answer 2

The purpose of the magnetic force is to change the direction of the velocity. The charged particle is accelerated because the direction changes and not because of a speed change.





This is how the lorentz force works. It wasn't derived. It was found empirically. The force must always be perpendicular to the velocity of the charged particle. But work is defined in terms of scaler products. i.e the direction of the force relative to the displacement must be taken into account. In this case having a perpendicular relation would mean that work is 0.









Likewise,

The motion of the particle along its trajectory (tangential velocity) will not change...ie..there is a 0 acceleration in the direction of motion. Ie. the particle will spin in a circule for ever unless it encouters another force ...like friction, or another columb force.



Lastly,

in the equation qvB = m VV /r



if you increase B the radius will get smaller. Nothing else will change. V will remain constant because it refers to the tangential velocity of the particle

Answer 3

The particle is accelerating: the trajectory is a circle.



There is no work associated with it as the force is normal to the displacement, hence the dot product of force and displacement = work is 0.



The speed (not velocity) v does not change, otherwise there would be work done and an increase in energy. The radius changes.

Answer 4

The particle cannot accelerate any more once its speed is equal to the pressure of the force. It's like a sail on a ship that cannot go any faster once the speed equals the combined sail pressure.