# Question 19cdd

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 $\vec{B}$ is

$\vec{F} = q \vec{v} \times \vec{B}$

Newton's second law of motion states that the force on the particle is equal to the rate of change of its momentum; $\therefore \vec{F} = \vec{\dot{p}}$. So we get
$\vec{\dot{p}} = q \vec{v} \times \vec{B}$
As momentum $\vec{p} = m \vec{v}$, if we take the dot product of both sides with $\vec{p}$. Since the vector $\vec{v} \times \vec{B}$ is perpendicular to both the vectors, the dot product of the RHS with $\vec{p}$ is zero.
=>vecp⋅vecdotp=0#
$\implies \frac{d}{\mathrm{dt}} | \vec{p} {|}^{2} = 0$
Dividing both sides with $2 m$, where $m$ is mass of the particle we get
$\frac{d}{\mathrm{dt}} | \vec{p} {|}^{2} / \left(2 m\right) = 0$
$\implies \frac{d}{\mathrm{dt}} \text{Kinetic Energy} = 0$

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