How to Show Force and Velocity are Perpendicular in a Magnetic Field?

[jlmac2001]

The force acting on a moving charge particle with mass m and charge q in a magnetic field B is the Lorentz force F= q(v x B), where v is it's velocity. Suppose thata particle moves in the (x,y) plane with a uniform B field in the z direction. Assuming Newton' second law, mdv/dt = F, show that the forceand velocity are perpendicular, and that both have constan magnitude. Hint: Find (d/dt)(v dot v).


I don't know where to start. How do you take the derivative of v dot v?

 

[HallsofIvy]

I suspect this will be moved to the "homework" section!

You differentiate v dot v using the "product rule":

The derivative of v dot v is: dv/dt dot v+ v dot dv/dt. Since the dot product of vetors is commutative, this is the same as 2v dot dv/dt (should remind you of the derivative of v²).

Since you are told that "the Lorentz force F= q(v x B)
"mdv/dt = F" becomes m dv/dt= q(v x B)

 

[modmans2ndcoming]

personaly

I think you should take the dot product first, then take the derivative...it is easier to see what is being done and the algebra is simpler.