in phase, hence the microwaves are perfectly
reflected. Macroscopically, this behaviour is
described by the complex dielectric constant ε(ω),
which is the square of the complex refractive index
(ε1 + iε2 = (n1 + in2)2).
The refractive index of many metals gives
reflectivities close to 100% at low frequencies.
The penetration depth of electromagnetic waves
of wavelength λ is given by
δ = λ/4πn2. (1)
For example, for microwaves with λ = 12.2 cm
incident on aluminium, δ ≈ 1.2 μm.
These are similar to skin depths, i.e. the
attenuation depths of alternating currents of
frequency ω in metals. (The relation between skin
depth and refractive index for small frequencies is
discussed, e.g., by Feynman [5].)
Generating microwaves in magnetrons
The most powerful microwaves produced by solid
state devices, such as used in cell phones, are far
too weak for cooking. Instead electron beams in
vacuum tubes under the combined effect of electric
and magnetic fields are made to follow curved
trajectories (the detailed mechanism for which is
described below). Most microwave ovens use
magnetrons. First invented in 1921 and strongly
improved around 1940, magnetrons allow either
continuous or pulsed microwave generation with
powers up to megawatts and frequencies between
1 and 40 GHz. Efficiencies are around 80% and
lifetimes about 5000 hours.
A cylindrical cathode is at the axis,
several millimetres from a hollow circular anode
(figure 2). Inside the anode there are a number
cavities designed to resonate at 2.45 GHz. A
voltage of several kV is applied between the
electrodes and a magnetic field is applied parallel
to the axis such that electric and magnetic fields
are perpendicular to each other.
Electrons ejected by the cathode accelerate
radially at first, but because of the magnetic field
they start to followcycloidal paths. If the magnetic
field is strong enough, the electrons cannot reach
the anode but form a rotating space charge. The
resonant cavities of the anode interact with the
electrons by either accelerating or decelerating
them. Finally this leads to electron bunches
which move around the cathode at microwave