Current in the middle of the Earth

What if the Earth's magnetic field were caused by a current loop in the mantle?

This can be a decent approximation because the Earth's magnetic field is roughly that of a dipole.

It's easy to look-up the Radius of the earth, and we can use a rough estimate of 0.23 gauss for the horizontal component of the magnetic field at 40 degrees latitude.

A simple expression for the magnetic field due to a dipole in spherical polar coordinates is


There's no dependence on azimuthal angle because of the symmetry axis along the dipole.

The dipole moment is then


And the current in a loop 1/3 the radius of the earth required to create a field of that strength is


That's a lot of statcoulombs!

This is Problem 2.2 from Classical Electromagnetic Radiation; Chapter 2 covers multi-pole expansions of static electric and magnetic fields.