Chapter 32, Problem 61P

SSMThe magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth’s center are given by

$B_h=\frac{{\mu }_0\mu }{4\pi r^3}\cos {\lambda }_m,B_v=\frac{{\mu }_0\mu }{2\pi r^3}\sin {\lambda }_m,$

where λ_m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth’s magnetic dipole moment has magnitude µ = 8.00 × 10²² A ž m². (a) Show that the magnitude of Earth’s field at latitude λ_m is given by

$B=\frac{{\mu }_0\mu }{4\pi r^3}\sqrt{1+3{\sin }^2{\lambda }_m}.$

(b) Show that the inclination ϕ_l of the magnetic field is related to the magnetic latitude λ_m by tan ϕ_l = 2 tan λ_m.