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Homework Help: Charged Particle in Magnetic Field

  1. Oct 19, 2005 #1
    We have a particle with electric charge [tex]e[/tex] that moves in a strong magnetic field [tex]B[/tex]. The particle is constrained to the (x,y)-plane, and the magnetic field is orthogonal to the plane, and constant with [tex]B[/tex] as the [tex]z[/tex]-component of [tex]\mathbf{B}[/tex].

    Furthermore we have the rotationally symmetric form of the vector potential,

    [tex]\mathbf{A} = -\frac{1}{2}\mathbf{r}\times \mathbf{B}[/tex]

    the relation between between velovity and momentum

    [tex]\mathbf{v} = \frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A} \right)[/tex]

    and the Hamiltonian

    [tex]H = \frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A} \right)^2[/tex]

    We now have to show by use of the equation of motion that, generally, the mechanical angular momentum

    [tex]L_{mek} = m(xv_y-yv_x)[/tex]

    is not a constant ot motion, whereas

    [tex]L_{mek}+\left(\frac{eB}{2c} \right)r^2[/tex]

    is conserved.

    I'm really not sure where or how to start. Do I have to use the Lagrangian and the Euler-Lagrange equation? If so, how do I find out wether or not any of these expressions are constants of motion or not? Or am I way off?

    Edit: minor correction to equations.
     
    Last edited: Oct 19, 2005
  2. jcsd
  3. Oct 19, 2005 #2

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    I don't know what you mean in your first equation by taking the cross product of 1/2 and B, but to answer your question, generally conserved quantities arise when the langrangian (or hamiltonian) is symmetric with respect to a change with one of the coordinates. Conservation of angular momentum comes from symmetry in rotations, ie, in the coordinate theta. Find a system of coordinates such that there is some theta-like coordinate with respect to which the lagrangian is invariant.
     
  4. Oct 19, 2005 #3
    First off, I've corrected the first equation, now it hopefully makes more sense :smile:

    Second, I'm afraid it didn't help me much, the rest of your post. You see, this is really a quantum mechanical course, so I'm a bit surprised we got a problem like this. Anyway, I don't know how to find "a system of coordinates such that there is some theta-like coordinate with respect to which the lagrangian is invariant", or even where to begin with all this. I don't think this is supposed to be such a difficult problem, but then again, everything is relative :wink:
     
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