Difference between revisions of "Forest UCM PnCP"

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Consider the impact on solving Newton's second law when there is an external Force that is velocity dependent
 
Consider the impact on solving Newton's second law when there is an external Force that is velocity dependent
  
:<math>\Sum \vec {F}_{ext} = \vec{F}(v) = m \frac{dv}{dt}</math>
+
:<math>\sum \vec {F}_{ext} = \vec{F}(v) = m \frac{dv}{dt}</math>
: \Rightarrow  \int_{v_i}{v_f} \frac{dv}{F(v) = \int_{t_i}{t_f} \frac{dt}{m}
+
: <math>\Rightarrow  \int_{v_i}{v_f} \frac{dv}{F(v) = \int_{t_i}{t_f} \frac{dt}{m}</math>
 
 
  
 
=Charged Particle in uniform B-Field=
 
=Charged Particle in uniform B-Field=

Revision as of 11:39, 29 August 2014

A Damping force that depends on velocity (F(v))

Newstons seconds law

Consider the impact on solving Newton's second law when there is an external Force that is velocity dependent

[math]\sum \vec {F}_{ext} = \vec{F}(v) = m \frac{dv}{dt}[/math]
[math]\Rightarrow \int_{v_i}{v_f} \frac{dv}{F(v) = \int_{t_i}{t_f} \frac{dt}{m}[/math]

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]\vec{v} = v_x \hat i + v_y \hat j[/math]
[math]\vec{B} = B \hat k[/math]


Lorentz Force
[math]\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]W = \Delta K.E.[/math]

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


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

Apply Newton's 2nd Law

[math]ma_x = qv_yB[/math]
[math]ma_y = -qv_x B[/math]
[math]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]\omega=\frac{qB}{m}[/math] = fundamental cyclotron frequency

Then we have two coupled equations

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

determine the velocity as a function of time

let

[math]v^* = v_x + i v_y[/math] = complex variable used to change variables
[math]\dot{v}^* = \dot{v}_x + i \dot{v}_y[/math]
[math]= \omega v_y + i (-\omega v_x)[/math]
[math]= -i \omega \left ( \omega v_x +i\omega v_y \right )[/math]
[math]= -i \omega v^*[/math]
[math]\Rightarrow[/math]
[math]v^* = Ae^{-i\omega t}[/math]

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

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

The above expression indicates that [math]v_x[/math] and [math]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]v_{\perp}[/math] such that

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

Determine the position as a function of time

To determine the position as a function of time we need to integrate the solution above for the velocity as a function of time

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

Using the same trick used to determine the velocity, define a position function using complex variable such that

[math]x^* = x + i y[/math]

Using the definitions of velocity

[math]x^* = \int v^* dt = \int v_{\perp}e^{-i\omega t} dt[/math]
[math]= \frac{v_{\perp}}{i \omega} e^{-i\omega t} [/math]

The position is also composed of two oscillating components that are out of phase by 90 degrees

[math]x^* = x + i y= \frac{v_{\perp}}{i \omega} e^{-i\omega t} = -i\frac{v_{perp}}{\omega} \left ( \cos(\omega t) - \sin(\omega t) \right )[/math]

The radius of the circular orbit is given by

[math]r = \left | x^* \right | = \frac{v_{perp}}{\omega} = \frac{mv_{perp}}{qB}[/math]
[math]r = \frac{p}{qB}[/math]
[math]p=qBr[/math]

The momentum is proportional to the charge, magnetic field, and radius


http://hep.physics.wayne.edu/~harr/courses/5200/f07/lecture10.htm


http://www.physics.sfsu.edu/~lea/courses/grad/motion.PDF

http://physics.ucsd.edu/students/courses/summer2009/session1/physics2b/CH29.pdf

http://cnx.org/contents/77faa148-866e-4e96-8d6e-1858487a520f@9

Forest_Ugrad_ClassicalMechanics