$($E$8$M$\mathrm{.}6)$ The aurora is caused by electrons and protons from the sun that get captured by the

earth$$s magnetic field and spiral into its atmosphere along the direction of the earth$$s

magnetic field. The typical speed of captured electrons is about $0\mathrm{.}003$c$,$ and the typical speed

for captured protons is about $20$ times smaller. What are the maximum radii of the circular

component of these particles$$ motions around the field direction? What are their orbital

frequencies?