With
the approximation of scaleindependent resistivity,

copacitative
time constant resistance ·
capacitance=constant 
(2.36) 
This implies that the
time required for a capacitor to discharge through a
resistor in a pure RC circuit is independent of
scale; with the scale dependence of the LR time
constant, however, a structure with fixed proportions can
change from a nearly pure RC circuit (if built
on a small scale) to a nearly pure LR circuit
(if built on a large scale). The nanometerscale RC
time constant indicated by this expression is (17
)×(9×10^{–21} F)
≈1.5
×10^{–19} s, but this result is nonphysical
because it neglects the effects of electron inertia and
relaxation time.
The LC product defines an oscillation
frequency

oscillation frequency
L^{1} 
(2.37) 
The characteristic
inductance and capacitance calculated above would yield
an LC circuit with an angular frequency of ~ 3
×10^{17} rad/s. Alternatively, in structures such
as waveguides,

oscillation frequency
L^{1} 
(2.38) 
To propagate in a
hypothetical waveguide 1 nm in diameter, an
electromagnetic wave would require a frequency of ~ 9
×10^{17} rad/s or more. Even the lower of the
two frequencies just mentioned corresponds to quanta with
an energy of ~ 30 aJ, that is, to photons in the xray
range with energies of ~ 200 eV. These frequencies and
energies are inconsistent with physical circuits and
waveguides (metals are transparent to xrays, electrons
are stripped from molecules at energies well below 200
eV, etc.). Quantum effects and electron inertia make Eq.
(2.38) inapplicable in the nanometer range.
Scale also affects the quality of an
oscillator:

Q oscillation frequency
L 
(2.39) 
Since Q is a measure of
the damping time relative to the oscillation time, small
AC circuits will be heavily damped unless nonclassical
effects intervene.
Where the following chapters consider
electromagnetic systems at all, they describe systems
with currents and fields that are slowly varying by the
relevant standards. Highfrequency quantum electronic
devices, though undoubtedly of great importance, are not
discussed here.
2.5. Scaling
of classical thermal systems
2.5.1.
Basic assumptions •
The classical continuum model assumes
that volumetric heat capacities and thermal
conductivities are independent of scale. Since heat flows
in nanomechanical systems are typically a side effect of
other physical processes, no independent assumptions are
made regarding their scaling laws.
2.5.2.
Major corrections
Classical, diffusive models for heat
flow in solids can break down in several ways. On
sufficiently small scales (which can be macroscopic for
crystals at low temperatures) thermal energy is
transferred ballistically by ›phonons for which the
mean free path would, in the absence of bounding
surfaces, exceed the dimensions of the structure in
question. In the ballistic transport regime, interfacial
properties analogous to optical reflectivity and
emissivity become significant. Radiative heat flow is
altered when the separation of surfaces becomes small
compared to the characteristic wavelength of blackbody
radiation, owing to coupling of nonradiative
electromagnetic modes in the surfaces. In gases,
separation of surfaces by less than a mean free path
again modifies conductivity. The following assumes
classical thermal diffusion, which can be a good
approximation for liquids and for solids of low thermal
conductivity, even on scales approaching the nanometer
range.
