Question

Earth's magnetic field varies in magnitude and direction from point to point in space. This spatial variation is characterized by the gradient and curvature of the field. Together, the field's gradient and curvature cause charged particles in the plasmasphere to experience a "longitudinal drift", resulting in an orbital motion around the Earth. Use the following formula (instead of textbook eq. 6.64) tau_d=2piqBr^2/3E_k, r=R_E+h, B=B_0(R/r)^3 to calculate the longitudinal drift orbital periods of 10 MeV protons at orbit altitudes of h=500 km and h=20,000 km, and compare your results to Table 6.5. E_k is proton kinetic energy.

Answer 1

Please provide the details of the symbols B₀ and R so that it becomes easy to understand the question. I researched for an equation of the form given in the question and it reqiures specification of the values(or atleast, names) of the symbols. Please update the same in the comments section and I'll upload a picture of the whole solution.

Edit 1: Although the difference between R and R_E is not very clear, we will take both the values as 6371000m (average radius of the earth). The value of B₀ is 31 $\mu T$ =31X106(-6) Tesla. q=1.6 X10^(-19) Coulomb. E_k=1.6X10^(-12) Joules.

Put all these given values in the three equations after which you will get your answer.