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} = q\left ( \begin{matrix} \hat i & \hat j & \hat k \\ v_x & v_y &0 \\ 0 &0 & B \end{matrix} \right ) }[/math]

[math]\displaystyle{ \vec{F} = q \left (v_y B \hat i - v_x B \hat j \right ) }[/math]

Apply Newton's 2nd Law

[math]\displaystyle{ ma_x = qv_yB }[/math]

[math]\displaystyle{ ma_y = -qv_x B }[/math]

[math]\displaystyle{ ma_z = 0 }[/math]

Motion in the z-direction has no acceleration and therefor constant (zero) velocity.

Motion in the x-y plane is circular

Let

[math]\displaystyle{ \omega=\frac{qB}{m} }[/math] = fundamental cyclotron frequency

Then we have two coupled equations

[math]\displaystyle{ \dot{v}_x = \omega v_y }[/math]

[math]\displaystyle{ \dot{v}_y = - \omega v_x }[/math]

determine the velocity as a function of time

let

[math]\displaystyle{ v^* = v_x + i v_y }[/math] = complex variable used to change variables

[math]\displaystyle{ \dot{v}^* = \dot{v}_x + i \dot{v}_y }[/math]

[math]\displaystyle{ = \omega v_y + i (-\omega v_x) }[/math]

[math]\displaystyle{ = -i \omega \left ( \omega v_x +i\omega v_y \right ) }[/math]

[math]\displaystyle{ = -i \omega v^* }[/math]

[math]\displaystyle{ \Rightarrow }[/math]

[math]\displaystyle{ v^* = Ae^{-i\omega t} }[/math]

the complex variable solution may be written in terms of [math]\displaystyle{ \sin }[/math] and [math]\displaystyle{ \cos }[/math]

[math]\displaystyle{ v_x +i v_y = A \left ( \cos(\omega t) - i \sin ( \omega t) \right ) }[/math]

The above expression indicates that [math]\displaystyle{ v_x }[/math] and [math]\displaystyle{ v_y }[/math] oscillate at the same frequency but are 90 degrees out of phase. This is characteristic of circular motion with a magnitude of [math]\displaystyle{ v_{\perp} }[/math] such that

[math]\displaystyle{ v^* = v_{\perp}e^{-i\omega t} }[/math]