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Accordingly, the efficiency of electrostatic motors decreases with decreasing scale. The absence of long conducting paths (like those in electromagnets) makes resistive losses smaller to begin with, however, and a detailed examination (Section 11.7) shows that efficiencies remain high in absolute terms for motors of submicron scale. The above relationships show that electromechanical systems cannot be scaled in the simple manner suggested for purely mechanical systems, even in the classical continuum approximation.

Electromagnets are far less attractive for nanoscale systems, since

  magnetic field L (2.31)

At a distance of 1 nm from a conductor carrying a current of 10 nA, the field strength is 2×10–6 T. The corresponding forces,

  magnetic force area (magnetic field)2 L4 (2.32)

are minute in nanoscale systems: two parallel, 1 nm long segments of conductor, separated by 1 nm and carrying 10 nA, interact with a force of 2×10–23 N. This is 14 orders of magnitude smaller than the strength of a typical covalent bond and 11 orders of magnitude smaller than the characteristic electrostatic force just calculated. Magnetic forces between nanoscale current elements are usually negligible. Magnetic fields generated by magnetic materials, in contrast, are independent of scale: forces, energies, and so forth follow the scaling laws described for constant-field electrostatic systems. Nanoscale current elements interacting with fixed magnetic fields can produce more significant (though still small) forces: a 1 nm long segment of conductor carrying a 10 nA current experiences a force of up to 10–17 N when immersed in a 1 T field.

The magnetic field energy of a nanoscale current element is small:

  magnetic energy volume (magnetic field)2 L5 (2.33)

The scaling of inductance can be derived from the above, but is independent of assumptions regarding the scaling of currents and magnetic field strengths:

  inductance L (2.34)

The inductance per nanometer of length for a fictitious solenoid with a 1 nm 2 cross sectional area and one turn per nanometer of length would be ~ 10–15 h.

2.4.4. Magnitudes and scaling: time-varying systems

In systems with time-varying currents and fields, skin-depth effects increase resistance at high frequencies; these effects complicate scaling relationships and are ignored here. The following simplified relationships are included chiefly to illustrate trends and magnitudes that preclude the scaling of classical AC circuits into the nanometer size regime.

For LR circuits,

  inductive time constant L2 (2.35)

Combining the characteristic 17 resistance and 10–15 h inductance calculated above yields an LR time constant of ~ 6×10 –17 s. This time constant is nonphysical: it is, for example, short compared to the electron °relaxation time in a typical metal at room temperature (~ 10–14 s). In reality, current decays more slowly because of electron inertia (which has effects broadly similar to those of inductance) and because of the related effect of finite electron relaxation time.

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