10. Motion of charged particles in complex fieldsSavchenko Formulas, chapter 10 of 14, 10 formulas

q, m

charge and mass of the particle

v

velocity perpendicular to the field

B

induction of the field

R

radius of the circle

The Lorentz force is perpendicular to the velocity and so only turns it, the particle moves on a circle whose centripetal force is $qvB$. The radius is proportional to the momentum, so measuring the radius in a known field gives the particle's momentum, and through the accelerating voltage one gets the mass spectrometer, where ions of different mass land on different diameters. The sense of rotation depends on the sign of the charge.

v

speed of the particle

U

accelerating voltage traversed

d

gap to be crossed across the field

A magnetic field does no work, so the speed in it is constant and only the direction changes. The entry speed comes from the accelerating voltage via $mv^2/2 = qU$. To cross a gap $d$ across the field the circle's diameter $2mv/qB$ must be at least $d$, hence the cut-off field of a magnetron or the smallest voltage.

L

angular momentum of the particle or ring about the axis

$\Phi$

magnetic flux through its orbit

q

charge

When the flux through an orbit changes, the vortex electric field $E\cdot2\pi r = -d\Phi/dt$ produces a torque $qEr$, and the change of angular momentum is $-q\,\Delta\Phi/2\pi$ however the field changed. So a charged ring spins up when a field is switched on, to angular velocity $qB/2m$, and so the rotation of a particle entering a region of different field along the axis changes, Busch's theorem. The same quantity underlies Larmor precession.

T

period of revolution

$\omega$

cyclotron frequency

q, m

charge and mass

The period $2\pi R/v$ with $R = mv/qB$ depends on neither the speed nor the radius, only on the ratio $q/m$ and the field. The cyclotron rests on this, its dee voltage alternating at the same frequency so that the particle gains $qU$ every half turn. The time to turn through any angle $\alpha$ is $\alpha m/qB$. A weak field imposed on an atom shifts the electron's orbital frequency by $eB/2m$, the Larmor precession.

$\alpha$

angle between the velocity and the field

R

radius of the helix

h

pitch of the helix

$v_\parallel, v_\perp$

velocity components along and across the field

The longitudinal velocity component is untouched by the field, the transverse one gives circling with radius $mv_\perp/qB$, and together they make a helix. The pitch is the longitudinal speed times the period, which does not depend on speed, so a beam of particles with one speed and a small spread of angles gathers on the axis again after one turn, which is magnetic focusing. A diverging beam with a small angle $\delta\alpha$ has there a width of order $R\,\delta\alpha^3$.

$v_\perp, v_\parallel$

velocity components across and along the field

B₁

field where the angle is $\alpha$

B₂

largest field, at the throat of the mirror

$\alpha_{\text{кр}}$

smallest angle to the field at which a particle is still reflected

In a field that changes slowly along a line the flux through a particle's orbit is conserved, so $BR^2$ and with it $v_\perp^2/B$ stay constant, while the total speed is always conserved since the magnetic force does no work. Moving into stronger field the particle turns longitudinal speed into transverse and is reflected where $B = B_1/\sin^2\alpha$. Particles at small angles pass through the throat, and the trapped fraction is set by the solid angle.

E, B

parallel electric and magnetic fields over a length $l$

v

speed of the particle along the axis

L

distance to the screen

y, z

deflections by the electric and by the magnetic field

Over a short region the transverse momenta from each field are computed separately, the electric one gives $eEl/v$, the magnetic one $eBl$, and the particle then flies straight. The ratio of the deflections gives the speed, and either one with a known speed the ratio $e/m$. With a spread of speeds the trace on the screen lies on a parabola $y \propto z^2$, from which Thomson measured $e/m$ and found isotopes. Written with $p$ instead of $mv$ the formulas hold at relativistic speeds too.

$\vec E, \vec B$

perpendicular electric and magnetic fields

$v_{\text{др}}$

drift velocity, the same for all charges

$y_{\max}$

extent of the cycloid across the drift

In a frame moving at $E/B$ across both fields the electric field vanishes and the particle simply circles, while in the laboratory its motion is a circle plus a uniform drift, a cycloid. The drift velocity depends on neither charge, nor mass, nor the sign of the charge. A particle entering at speed $v$ circles in the drifting frame at $v - E/B$ and strays from the straight line by twice that circle's radius.

$\vec F$

constant force across the field, gravity, electric, any

q

charge, the drift due to gravity depends on its sign

$\alpha$

relative gradient of the field

Any constant force across a magnetic field gives a drift perpendicular to both the force and the field at speed $F/qB$, at which the Lorentz force of the drift just cancels $F$, the rest of the motion being a rotation. An electric force gives $E/B$ for all, gravity gives $mg/qB$, different in size and direction for ions and electrons, hence currents. A field gradient gives a drift too, the radius is smaller on the strong side and the particle shifts every turn.

E

electric field between cathode and anode

B

magnetic field along the plates

d

distance to the anode

$U_{\text{зап}}$

voltage below which electrons no longer reach the anode

An electron starting from rest moves on a cycloid drifting at $E/B$ and rises above the cathode no higher than $2mE/qB^2$, twice the radius of circling at speed $E/B$. The equations of motion integrate once to $mv_x = eBy$, and with energy conservation this gives the height without solving the full problem. When the height is less than the gap the anode current stops, which measures $e/m$ from the cut-off voltage or field.