Question #19cdd

1 Answer
Dec 23, 2016

We know that magnetic field can change the direction of the motion of a charged particles, but it will not change its speed*. Similarly, direction of momentum changes but magnitude of momentum does not change.

However, change in direction of velocity means that the velocity is changing. Therefore, linear momentum is not a constant in circular motion.

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Mathematically stated, Lorentz equation for magnetic part of force on a charge #q# moving with velocity #v# in a magnetic field #vecB# is

#vecF=qvecvxxvecB#

Newton's second law of motion states that the force on the particle is equal to the rate of change of its momentum; #:.vecF=vecdotp#. So we get
#vecdotp=qvecvxxvecB#
As momentum #vecp=mvecv#, if we take the dot product of both sides with #vecp#. Since the vector #vecvxxvecB# is perpendicular to both the vectors, the dot product of the RHS with #vecp# is zero.
#=>vecp⋅vecdotp=0#
#=>d/dt|vecp|^2=0#
Dividing both sides with #2m#, where #m# is mass of the particle we get
#d/dt|vecp|^2/(2m)=0#
#=>d/dt"Kinetic Energy"=0#

This means that kinetic energy is constant in time for such a motion. This is outcome of fact* stated above.