Motion in Crossed Electric and Magnetic Fields

Consider a particle of mass $m$ and electric charge $q$ moving in the uniform electric and magnetic fields, ${\bf E}$ and ${\bf B}$. Suppose that the fields are ``crossed'' (i.e., perpendicular to one another), so that ${\bf E}\cdot{\bf B} = 0$.

The force acting on the particle is given by the familiar Lorentz law:

$${\bf f} = q\,\left({\bf E} + {\bf v}\times{\bf B}\right),$$ (269)

where ${\bf v}$ is the particle's instantaneous velocity. Hence, from Newton's second law, the particle's equation of motion can be written

$$m\,\frac{d{\bf v}}{dt} = q\,\left({\bf E} + {\bf v}\times{\bf B}\right).$$ (270)

It turns out that we can eliminate the electric field from the above equation by transforming to a different inertial frame. Thus, writing

$${\bf v} = \frac{{\bf E}\times{\bf B}}{B^2} + {\bf v}',$$ (271)

Eq. (270) reduces to

$$m\,\frac{d{\bf v}'}{dt} = q\,{\bf v}'\times{\bf B},$$ (272)

where we have made use of the fact that ${\bf E}\cdot{\bf B} = 0$. Thus, we conclude that the addition of an electric field perpendicular to a given magnetic field simply causes all particles, irrespective of their charge or mass, to drift perpendicular to both the electric and magnetic field with the velocity

$${\bf v}_{EB} = \frac{{\bf E}\times{\bf B}}{B^2}.$$ (273)

Hence, the electric field has no effect on particle motion in a frame of reference which is co-moving with the so-called E-cross-B velocity given above.

Let us suppose that the magnetic field is directed along the $z$-axis. As we have just seen, in the ${\bf E}\times{\bf B}$ frame, the particle's equation of motion reduces to Eq. (272), which can be written:

$$\frac{d v_x'}{dt} = {\mit\Omega}\,v_y',$$ (274)

$$\frac{d v_y'}{dt} = -{\mit\Omega}\,v_x',$$ (275)

$$v_z' = 0.$$ (276)

Here,

$${\mit\Omega} = \frac{q\,B}{m}$$ (277)

is the so-called cyclotron frequency. Equations (274)-(276) can be integrated to give

$$v_x' = v_\perp\,\sin({\mit\Omega}\,t),$$ (278)

$$v_y' = v_\perp\,\cos({\mit\Omega}\,t)$$ (279)

$$v_z' = v_\parallel,$$ (280)

where we have judiciously chosen the origin of time so as to eliminate any phase offset in the arguments of the above trigonometrical functions. According to Eqs. (278)-(280), in the ${\bf E}\times{\bf B}$ frame, charged particles gyrate at the cyclotron frequency in the plane perpendicular to the magnetic field with some fixed speed $v_\perp$, and stream parallel to the magnetic field with some fixed speed $v_\parallel$. The fact that the cyclotron frequency is positive for positively charged particles, and negative for negatively charged particles, just means that oppositely charged particles gyrate in opposite directions in the plane perpendicular to the magnetic field.

Equations (278)-(280) can be integrated to give

$$x' = -{\mit\rho}\,\cos({\mit\Omega}\,t),$$ (281)

$$y' = {\mit\rho}\,\sin({\mit\Omega}\,t)$$ (282)

$$z' = v_\parallel\,t,$$ (283)

where we have judiciously chosen the origin of our coordinate system so as to eliminate any constant offsets in the above equations. Here,

$${\bf\rho} = \frac{v_\perp}{{\mit\Omega}}$$ (284)

is called the Larmor radius. Equations (281)-(283) are the equations of a spiral of radius ${\mit\rho}$, aligned along the direction of the magnetic field (i.e., the $z$-direction).

Hence, we conclude that the general motion of a charged particle in crossed electric and magnetic field is a combination of ${\bf E}\times{\bf B}$ drift [see Eq. (273)] and spiral motion aligned along the direction of the magnetic field. Particles drift parallel to the magnetic field with constant speeds, and gyrate at the cyclotron frequency in the plane perpendicular to the magnetic field with constant speeds. Oppositely charged particles gyrate in opposite directions.