Question #19cdd

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.