Gyromotion of a charged particle in a magnetic field

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A charged particle of mass $m$ and charge $q$ moving with a velocity $\boldsymbol{v}$ in an an electric field $\boldsymbol{E}$ and a magnetic field $\boldsymbol{B}$ is subject to a Lorentz force, $\boldsymbol{F}$, given by $$ \boldsymbol{F} = q(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}). $$ The equation of motion for a single particle is therefore given by Newton's second law as $$ \boldsymbol{\ddot{r}} = \frac{q}{m}(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}). $$ Here we will consider a uniform magnetic field, $\boldsymbol{B} = (0,0,B)$ and zero electric field, $E=0$. In this case, the trajectory of the particle can be obtained by solving the equation of motion analytically, but here we integrate it numerically using SciPy's integrate.odeint method. Assuming the particle starts off with non-zero components of its velocity parallel ($v_\parallel$) and perpendicular ($v_\perp$) to the magnetic field, it moves in a helix, with radius given by $$ \rho = \frac{mv_\perp}{|q|B}, $$ known as the Larmor or cyclotron radius (or gyroradius).

With the numerical approach the code is straightforward, and given in a Jupyter Notebook which is also hosted on my Github page. The motion of an electron and an ion are represented qualitatively: because the mass ratio between these particles is at least $\mu = m_e/m_p = 1836$, we compare only particles differing in mass by a factor of 3. Matplotlib's animation.FuncAnimation method is used to produce an animated movie of the motion.

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AndrewL 2 months, 1 week ago

Greate job, thanks a lot!

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