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.
- [math]\displaystyle{ \vec{v} = v_x \hat i + v_y \hat j }[/math]
- [math]\displaystyle{ \vec{B} = B \hat k }[/math]
- Lorentz Force
- [math]\displaystyle{ \vec{F} = q \vec{E} + q\vec{v} \times \vec{B} }[/math]
- Note
- the work done by a magnetic field is zero if the particle's kinetic energy (mass and velocity) don't change.
- [math]\displaystyle{ W = \Delta K.E. }[/math]
No work is done on a charged particle force to move in a fixed circular orbit by a magnetic field (cyclotron)
- [math]\displaystyle{ \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 ) }[/math]