Through a Glass Magnetically

About a month ago, my wonderful fiancee bought us each an Apple watch, so we can keep track of our fitness goals.  To make the watch water-proof, the designers avoided any external ports for charging, which means the watch uses a type of wireless power transfer, which I've mentioned before.

The charging pad consists of a coil of wire with current running in a spiral.  The power is transferred thanks to the Maxwell-Faraday law:

The "∇ ×" is called the curl operator, and quantifies how much the current is going in a circle. The equation says that anytime you have an electric field going in a circle, you get a changing magnetic field in the middle.  This magnetic field can carry energy into another coil of wire in the watch, but there's a problem: The law applies to any material the field moves through.

Metal is a conductor, which means it has electrons that are free to move.  If the field passes by them, they'll start moving in a circle too, and absorb some of the energy. The amount the field is absorbed is quantified by the skin depth:

where ρ measures how easily the electrons move in the conductor, ω is the frequency of the electromagnetic wave, and μ tells how magnetic fields behave in the material.  Wireless power is usually transferred with a frequency on the order of 5 kHz, which for aluminum gives a depth of about 1 mm.  That may seem like a reasonable amount, but the effect is exponential – For every 1 mm, the efficiency drops by about 37%.

To avoid wasting power, Apple used a disk of glass on the back of the watch, allowing the waves to pass freely:

Initially, I had assumed the glass was purely for the heart rate sensor, but after reading the rumors that the next iPhone may have wireless charging, and therefore a glass case, I realized the true purpose.