3. (a) The Lorentz force on a charged particle is F=qE + qv x B. In the absence of an electric field, show that a charged particle's motion can be resolved into two components: one constant, along the magnetic field, and one periodic, perpendicular to the magnetic field. 3. (b) Derive the relations for the gyrofrequency and gyroradius: qB m Re - mv aB 3. (c) Calculate the gyrofrequency (in hertz) of the following: proton in a 100-nT field electron in a 1,000-nT field (iii) singly-ionized oxygen atom in a 50,000-nT field. 3. (d) Assume the Earth's magnetic field (in the equatorial plane) is given by B = 31 x 10 nT (R /r) where R - 6380 km where r is radial equatorial distance (geocentric). (i) Calculate at (roughly) what distances from the Earth would the gyrofrequencies of part (c) be found in the equatorial plane. (ii) What is the gyroradius of a proton moving transverse to a 100-nT field at 2x10' meters per sec? How does this compare with the distance (near the equator) over which the Earth's dipole field changes by a factor of 2 near the region where B #100 nT? Would you expect the proton to conserve its first adiabatic invariant