Basics of Medical Physics (Ukázka, strana 99)

4.

BIOPHYSICS ELECTRIC PHENOMENA OF

Bioelectric phenomena include cell membrane potentials which are of particular importance in nerve cells where they produce action potentials, include heart electrical phenomena such as the natural pacemaker which triggers the heart electrical sequence, and include other bioelectric measurements.

4.1 INTRODUCTION Electric biosignals are generated by nerve cells and muscle cells. Its source is the membrane potential, which under certain conditions may be excited to generate an action potential.

In single cell measurements, the action potential itself is the biomedical signal. The electric field propagates through the biologic medium, and thus the potential may be acquired at relatively convenient locations on the surface. Electrical biosignals are usually taken to be electric currents produced by the sum of electrical potential differences across a specialized tissue, organ or cell system.

Electric charge is present in all bodies of our surroundings and also in our body. It is one of the basic physical properties of elementary particles and it belongs to the basic physical quantities. The charge is positive or negative. The quantum of charge is the charge of one electron or of one proton, i.e. 1.6×10−19 (C). The unit of electric charge is 1 coulomb (C). Since the number of particle present in 1 mole is given by Avogadro’s constant NA, total charge of 1 mole of univalent ions is so called Faraday’s constant F = e.NA= 1.6×10−19(C)×6.02×1023(mol−1) = = 96.484 kC.mol−1.

In any system the law of conservation of electric charge holds: During any process, the net electric charge of an isolated system remains constant (is conserved).

4.1.1 Coulomb law and permittivity Coulomb law quantitatively describes the interaction among electric charges. Attractive or repulsive force F acting between two charges, q0 and q is directly proportional to their product and inversely proportional to the squared distance r between them.

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F=

1 q0 q .

4πε r 2

(4.1)

The constant ε is called permittivity and its value for vacuum is ε0 = 8.85×10−12 F.m−1.

Permitivity is a measure of how an electric field affects, and is affected by, a dielectric medium. Dielectric polarisation is observed in insulators and thus the force between the charges is decreased as compared with that in vacuum. Therefore, permittivity value in insulators is higher than ε0. The relative permittivity εrel = ε/ε0 is usually tabled. Relative permittivity is the ratio of the capacitance of a capacitor using that material as a dielectric, compared to a capacitor that has a vacuum as its dielectric. For illustration, some values are presented in the following table.

Tab. 4.1 Relative permittivity in some media Medium

ε/ε0

Medium

ε/ε0

Vacuum

1

Cell membrane

8

Air

1.0006

Water

78.5

High value of relative permittivity of water enables very good solubility of salts in water.

In water, ions may be distanced since water molecules decrease the attractive forces between negatively and positively charged ions. From the energy point of view, hydration of ions is accompanied by a decrease of free enthalpy (Gibbs energy). Various ions are hydrated in various extents. The measure of hydration is given by the number of water molecules that have lost their translation degrees of freedom due to the interaction with the given ion. The positively charged ions possess a higher hydration number since positive charge manifests a higher polarisation effect on electron shells in water molecule as compared with negative charge. The ions of smaller size possess a higher hydration number than larger ones of the same charge. Therefore, the order of relative size of hydrated K+ and Na+ ions in water solution is reversed in comparison with the order of their radii in crystalline lattice. That is why the cell membranes are more permeable for K+ than for Na+ ions. This fact plays its role at the transport of ions across the membrane and at the formation of membrane potential.

Tab. 4.2 Radius of some ions (nm) Radius of ion in crystalline lattice

Effective radius in water solution

Na+

0.095

0.24

K

0.133

0.17

+

The electric field that exists at a point is the electrostatic force F experienced by a small test charge q0 placed at this point divided by the charge itself. It is a vector. The intensity of electric field E is the force per unit charge. Therefore, its unit is N/C. It can be also expressed by using the unit of potential, volt. Since 1 J = 1C×1V, then N/C = N.m/C.m = J/(C.m) = V/m.

The intensity of electric field is given by E=

1 q F = . (4.2) q0 4πε 0 r 2 99

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Direction of electric field intensity is shown with field map lines. In general, it directed from positive charge to negative.

4.1.2 Electric potential, potentials of phase boundary-lines Tab. 4.3 Review of symbols applied for various potentials and their explanation Symbol and quantity

Explanation

V Electric potential

Physical quantity expressed in volts. Zero potential is theoretically considered in an infinite distance from conductor, practically in such distance when the action of electrostatic forces can be neglected. In practice, potential of Earth is considered as zero potential.

φ Internal (Galvani) potential

φ = ψ + χ; internal potential of the given phase (e.g. solid phase) in the medium of other phase (e.g. liquid phase)

Ψ External (Volt) potential

The work required for the transport of unite charge from infinity to the distance of about 10−6 cm from the surface of the given phase (to win electrostatic forces)

χ Surface potential difference

The work required for the transport of unite charge across the surface

µ i = Electrochemical potential

µ i = µi + zi Fφ ; The work required for transport of 1 mole of the i-th component (ion or electron) inside the given phase, defined as sum of chemical and electrostatic components (F = 96 484 C/mol – ­Faraday’s constant, zi – number of elementary charges of the i-th ion).

μi Chemical potential of the i-th component

µi = µi0 + RT .ln ai ; Partial molar Gibbs energy, i.e. the work corresponding to the change of the i-th component by 1 mole (R – universal gas constant, T – absolute temperature, ai – activity of the i-th component).

E Electrode potential

Potential on electrode surface. Potential of standard hydrogen electrode is usually considered as zero potential.

The electric potential V at a given point is the electric potential energy EPE of a small test charge situated at this point divided by the charge itself. Its unit is volt (V), and it holds volt = joule/coulomb.

Potential of a point charge is given by V=

1 q EPE = . (4.3) 4πε 0 r q0

Let us consider electric field around charged conductor. Let us assume that the charge q of the same sign is transported from infinity to the place of the conductor. The work W must be done to win the repulsive force F. This work equals the potential energy of the charge q.

Potential energy of the unite charge depends on the position of the charge in the field. It is called the potential at the given place.

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