Forest UCM PnCP

Charged Particle in uniform B-Field

Consider a charged particle moving the x-y plane in the presence of a uniform magnetic field with field lines in the z-dierection.

$\vec{v} = v_x \hat i + v_y \hat j$

$\vec{B} = B \hat k$

Lorentz Force

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

Note

the work done by a magnetic field is zero if the particle's kinetic energy (mass and velocity) don't change.

$W = \Delta K.E.$

No work is done on a charged particle force to move in a fixed circular orbit by a magnetic field (cyclotron)

$\vec{F} = m \vec{a} = q \vec{v} \times \vec{B} = \left ( \begin{matrix} \hat i & \hat j & \hat k \\ v_x & v_y &0 \\ 0 &0 & B \end{matrix} \right )$

Forest_Ugrad_ClassicalMechanics