# Blog

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## ExB drift for an arbitrary electric potential

As described in the previous blog post charged particle moving in crossed constant magnetic and electric fields exhibits a drift velocity, $(\boldsymbol{E}\times\boldsymbol{B})/B^2$, perpendicular to both $\boldsymbol{E}$ and $\boldsymbol{B}$. The particle's trajectory in this situation can be found analytically. For an arbitrary $\boldsymbol{E}$, some kind of numerical integration of the equation of motion is usually necessary, but the force that the particle experiences at an instant is perpendicular to to the electric field and the particle therefore undergoes its gyromotion along an isocontour of electrostatic potential, $V$ (since $\boldsymbol{E} = -\nabla V$).

## Depicting a torus as an SVG image

SVG may not be the most obvious choice for depicting a 3-D object, but with some care over the perspective and ordering of the plotted points, it can be done.

## Reaching Orbit

The following code illustrates the effect of the initial velocity on the dynamics of an object released in a gravitational field. A very simple numerical integration of the equation of motion gives the trajectory, which is plotted below for four different initial speeds for a rocket released at 200 km altitude parallel to the Earth's surface. At this altitude the speed needed for a circular orbit is about 7.8 km/s. You can read more about this kind of simulation at the Wikipedia page for Newton's cannonball.

## Moiré patterns in a pair of hexagonal lattices

A Moiré pattern is an interference pattern that occurs when two grids of repeating lines or shapes are rotated by a small amount relative to one another (oblig. xkcd).

## Gyromotion of a charged particle in a magnetic field

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).