© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

4.7 Electric and Magnetic Sensing

4.7.1 Electric Fields

The electric field E (volts/m) caused by a single point charge q (e.g., an electron or singly-charged ion) is given by Coulomb's law as

{Eqn. 4.40}

where q_e = 1.60 x 10^-19 coul (one charge), e₀ = 8.85 x 10^-12 farad/m (permittivity constant), k_e = dielectric constant of the matter traversed by the electric field (k_e = 74.31 for pure water at 310 K, 5.7 for diamond, 1 for vacuum or air), and r = distance from the charge, in meters. The field at a distance of 1 nm from a point charge is ~10⁹ volts/m in air, 2 x 10⁷ volts/m in water. The force between two unit charges F = q_e E = 230 pN in air, 3 pN in water, at 1-nm separation.

What is the magnitude of electric fields likely to be encountered in nanomedical situations? Electric fields in the human body may be generated from internal or external sources. Internally, ions and individual molecules may carry static electric fields. For example, the surface of an isolated charged amino acid (Section 3.5.5) has ~1 charge/nm². Debye-Huckel shielding due to counterion flow in salty fluids (as are found in the human body) reduces these fields very rapidly with distance. From Eqn. 3.20, the static field from a single charged amino acid floating in human plasma falls from ~6 x 10⁶ volts/m at a range of 1 nm to 0.7 volts/m at r = 10 nm. In plant cells, photosynthetic "receivers" detect gradients in the electrochemical potential of the order of 1 eV across distances on the order of 1 nm, an instantaneous ~10⁹ volts/m.

The most important internal electrical sources are the electrochemical gradients caused by gated channel and transporter molecular pump operations at the intracellular level; muscular, membrane, digestive, and neural activity at the intercellular and organ level (see also Section 4.9.3.1); and piezoelectric fields generated by movement of collagenous tissues (e.g., tendon and elastin), bones, and many other biological materials.^{1939­1942,3089­3095} Typically these sources generate potentials of 10-100 millivolts over distances of 0.01-10 microns, producing local fields in the range of 10³-10⁷ volts/m. For example, the 50 millivolt transmembrane potential necessary to open a sodium channel corresponds to an electric field of 5 x 10⁶ volts/m. (See also Section 4.8.7.) Interestingly, implanted electrical devices such as pacemakers may produce up to 600 volts/m one millimeter from the casing.

External sources may also contribute to detectable nanomedical electric fields within the human body, though usually to a lesser degree. Placing a hand flat between two plates charged to E_p = 3 x 10⁶ volts/m (near the breakdown voltage for air) induces an instantaneous field of E ~ E_p / k_water = 42,000 volts/m transversely through the limb. (A DC field induces ion current flows which accumulate surface charge and cancel out the internal field after ~one RC time.) Electric fields directly beneath high-tension power lines are 2,000-11,000 volts/m, or 50-1000 volts/m at a lateral displacement of 25 meters along the ground.⁴⁷⁷ Household appliances produce 200-300 volt/m fields;⁴⁷⁷ 20 amperes flowing through standard Romex copper wiring in the walls of a house produces 100-600 volts/m in the middle of a room, up to 24,000 volts/m at the wall (E = B / m₀ e₀ c; c = the speed of light, B from Eqn. 4.44); static sparks from carpet walking are ~10,000 volts/m, and ~10⁵ volts/m makes hair stand on end. Quiescent atmospheric charge is ~100 volts/m, rising to 5000 volts/m during severe thunder-storms oscillating from positive to negative over a ~10,000 volt/m range. Lightning bolts (~20 coul 0.002-sec discharge) produce 60,000 volt/m spikes at a distance of 10 meters from the strike, 600 volts/m at a range of 1 kilometer. Electron-hole avalanches in semiconductors are initiated by fields >~5 x 10⁷ volts/m.¹²⁹ Penal electrocutions use ~1500 volts/m. (See also Section 7.2.3.)

Direct force sensing of these electric fields is ineffective; from Eqn. 4.40, the minimum detectable field E_min = F_min / q = 6 x 10⁷ volts/m using F_min = 10 pN (Section 4.4.1) and a single-charge sensor. Particle deflection seems more efficient. Consider a collimated stream of singly-charged particles, each of mass m and initial velocity V₀ entering an evacuated test chamber of length L and experiencing a uniform electric field E, causing a lateral displacement Dx = q E L²/ 2 m V₀² measured by the sensor. For this measurement, SNR ~ln (m V₀² / 2 kT), so the minimum detectable electric field is

{Eqn. 4.41}

For Dx_min = 10 pm (Section 4.3.1), T = 310 K and SNR = 2, E_min = 10,000 volts/m for a sensor of size L = 28 nm, or 100 volts/m for L = 280 nm, allowing measurement of all nanomedically relevant fields that are homogeneous across the entire device. Using the larger 280-nm sensor, time of flight is ~10^-12 sec for electrons or ~10^-9 sec for U²³⁸ ions, permitting t_meas ~ 10^-9 sec and dissipating at least ~30 pW waste heat (due to kinetic impact) in continuous operation. Time-varying electric fields can be measured up to t_meas^-1 ~ GHz frequencies, and electric current may be determined using standard techniques because current density is scale-invariant.¹⁰ Care must be taken to eliminate measurement error due to exogenous magnetic fields. Piezoelectric or other electromechanical transducers can convert electricity into mechanical signals (Section 6.3.5) appropriate for all-mechanical computation and control systems.

Another class of electric field detector of similar size and sensitivity is the electrosensitive gel sensor. For example, the negatively charged heparan sulfate proteoglycan network found in the secretory granule matrix expands 50% in volume within milliseconds of exposure to a -2 volt/micron field, and the response appears linear with voltage.⁴⁹⁸ Thus the length of a 400-nm long gel-filled sensor of fixed cross-sectional area should expand a minimally-detectable 10 pm (= Dx_min) in a 100 volt/m field. Similar sensitivities to small periodic fields, analogous to shark electroreceptors (maximum sensitivity ~10^-7 volts/m,⁸¹³), are found in ion-channel embedded artificial membranes that selectively amplify electric signals in noisy environments via stochastic resonance.⁵¹⁴ Stochastic resonance in neurons permits detection of electric fields as weak as 10 volts/m.⁵⁶⁶

The theoretical limits have been analyzed by Weaver and Astumian⁸¹⁴ and by Block,⁸¹⁰ who conclude that the minimum detectable static field for a hollow spherical sensor of radius r_sen and shell membrane thickness d_mem (capacitance C_e = 4 p e₀ k_e r_sen² / d_mem) is

{Eqn. 4.42}

Using k_e = 74.31 for water at T = 310 K, r_sen = 0.5 micron and d_mem = 10 nm gives E_minDC = 200 volt/m. Increasing r_sen to 5 microns reduces E_min to ~2 volts/m. To detect a periodic field of frequency n_e by sampling over (n_e t_meas) cycles for a measurement time t_meas, Block⁸¹⁰ gives

{Eqn. 4.43}

Using n_e = 1 MHz and t_meas = 1 sec gives E_minAC = 0.2 volt/m. Cylindrical sensors are 10-100 times more sensitive, and sensitivity may possibly be further enhanced by coupling the periodic electric potential to a Michaelis-Menten type enzyme or a similar nanomechanism embedded in the sensor shell to achieve tuning.⁸¹⁴ Fields as small as ~10^-4 volts/m may be detectable by ~20-micron mammalian living cells;⁸¹⁴ such detection may trigger biochemical observables which may be monitored or eavesdropped by resident nanomedical devices. Submicron-scale electrometers capable of ~0.1 electrons/Hz^1/2 (charge per unit bandwidth) have been demonstrated experimentally, and also single-electron transistors (SETs) to 10^-4 electron/Hz^1/2, with the potential to reach 10^-6 electron/Hz^1/2 .²⁹²⁶ Galvanotropic bacteria have been reported in the literature.³³⁵¹

Radiofrequency receivers for power transmission and communication in vivo are described in Sections 6.4.2 and 7.2.3. Antenna radiation patterns may also be useful in diagnostics and macrosensing applications (Section 7.2.3). Piezoelectric macrosensing of bone loads is briefly described in Section 4.9.3.3.

Last updated on 17 February 2003