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B Lorentz force

  1. Jul 16, 2017 #1
    We know that Lorentz Force is a centripetal force so:
    $$qVBsenx=m\frac{v^2}{r}$$ and the value of V is constant (but not the direction)
    But if we write:
    $$qvBsenx=m\frac{dV}{dt}$$
    we obtain that v is exponential.
    What is wrong ?
     
  2. jcsd
  3. Jul 16, 2017 #2

    vanhees71

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    I guess you want to solve the nonrelativistic equation of motion for a particle in a constant magnetic field. Let's put ##\vec{B}=B \vec{e}_z##. Then the equations of motion read (in SI units)
    $$m \frac{\mathrm{d} \vec{v}}{\mathrm{d}t} = q \vec{v} \times \vec{B}.$$
    In components you have
    $$\begin{pmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \end{pmatrix}=\frac{qB}{m} \begin{pmatrix} v_y \\ -v_x \\ 0 \end{pmatrix}=\omega \begin{pmatrix} v_y \\ -v_x \\ 0\end{pmatrix}.$$
    Now you can integrate the equations of motion quite easily. I give some hints:

    The equation for ##v_z## is very simple. This you should be able to solve very easily.

    To solve also for ##v_x## and ##v_y##, just take the time derivative of the equation for ##v_x## and use the 2nd equation to eliminate ##v_y##. Then you have a equation for ##v_x## alone, which is quite easy to solve too.

    Once you have ##\vec{v}## just take one more integration to get ##\vec{x}##.
     
  4. Jul 16, 2017 #3

    Orodruin

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    You are trying to write a vector equation as a single scalar equation.
     
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