Sparkle Plenty

I just had a pretty neat problem on my Electrodynamics (course text) mid-term. The problem is to solve for the electric field for two charged conducting spheres of different radii connected by a conducting filament so they reach the same potential.

The result (pdf of full solution) is that the electric field is greater near the surface of the smaller sphere even though it carries less of the total charge. The physical implications of this is that the electric field will be high near the parts of a charged conductor that have a small radius of curvature (ie corners). That's why you get arcing and sparking on forks and crumpled aluminum foil that you put in the microwave.

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.