Cyclotron Frequency The Lorentz force density is f E J B

Cyclotron Frequency: The Lorentz force density is ¯f = E¯ + J¯ × B¯ and has units [N/m3 ]. When the charge and current consist of a single charged particle, such as an electron, the charge density and current densities are delta functions in space. The force in Newtons on the charge q moving at velocity ¯v is F¯ = q E¯ + ¯v × B¯ . Consider a particle with charge q and mass m moving with speed v in a uniform static magnetic field in the zˆ-direction, B¯ = zBˆ 0. The Lorentz force is perpendicular to the direction of the velocity and the charge particle moves in the x-y plane. Let ¯v = ˆxvx + ˆyvy.

a. Show that F¯ = xqv ˆ yB0 + ˆyqvxB0.

b. Using Newton’s Law F¯ = mdv/dt ¯ , show that m vx t = qvyB0 m vy t = qvxB0.

c. Define the variable c qB0/m and use substitution to obtain the following differential 1 equations. 2vx t2 + 2 c vx = 0 2vy t2 + 2 c vy = 0.

d. The solutions to these equations are vx(t) = x(t) t = v cos(ct) vy(t) = y(t) t = v sin(ct), where v is the magnitude of the velocity. Let ¯v(t = 0) = vyˆ, for example. Integrate and find the equations for x(t) and y(t). For simplicity, let x(0) = v/c and y(0) = 0.

e. What shape does the charged particle trace in time? The frequency at which it repeats is the cyclotron frequency c = qB0/m.

Field and Waves

Solution

Electron cyclotron resonance is a phenomenon observed in plasma physics, condensed matter physics, and accelerator physics. An electron in a static and uniform magnetic field will move in a circle due to the Lorentz force. The circular motion may be superimposed with a uniform axial motion, resulting in a helix, or with a uniform motion perpendicular to the field, e.g., in the presence of an electrical or gravitational field, resulting in a cycloid. The angular frequency ( = 2f ) of this cyclotron motion for a given magnetic field strength B is given (in SI units)[1] by

   
     
       
         
         
            c
            e
         
       
        =
       
         
           
              e
              B
           
            m
         
       
     
   
    {\\displaystyle \\omega _{ce}={\\frac {eB}{m}}}

.

where

   
     
        e
     
   
    {\\displaystyle e}

is the elementary charge and

   
     
        m
     
   
    {\\displaystyle m}

is the mass of the electron. For the commonly used microwave frequency 2.45 GHz and the bare electron charge and mass, the resonance condition is met when B = 875 G = 0.0875 T. For particles of charge q, rest mass m0 moving at relativistic speeds v, the formula needs to be adjusted according to the special theory of relativity to:

   
     
       
         
         
            c
            e
         
       
        =
       
         
           
              q
              B
           
           
             
             
             
                m
               
                  0
               
             
           
         
       
     
   
    {\\displaystyle \\omega _{ce}={\\frac {qB}{\\gamma \\cdot m_{0}}}}

where

   
     
       
        =
       
         
            1
           
              1
             
              (
              v
             
                /
             
              c
             
                )
               
                  2
               
             
           
         
       
     
   
    {\\displaystyle \\gamma ={\\frac {1}{\\sqrt {1-(v/c)^{2}}}}}

.

An ionized plasma may be efficiently produced or heated by superimposing a static magnetic field and a high-frequency electromagnetic field at the electron cyclotron resonance frequency. In the toroidal magnetic fields used in magnetic fusion energy research, the magnetic field decreases with the major radius, so the location of the power deposition can be controlled within about a centimeter. Furthermore, the heating power can be rapidly modulated and is deposited directly into the electrons. These properties make electron cyclotron heating a very valuable research tool for energy transport studies. In addition to heating, electron cyclotron waves can be used to drive current.