Home :: Chapter 7 :: Web Topic 7.3

# Web Topic 7.3A primer on electrical signals

## Basic electrostatics

• Charge: An object (atom, molecule, piece of material containing many molecules) with unequal total numbers of electrons and protons is said to be charged: each excess electron adds a charge of -1 and each missing electron (excess proton) adds a charge of +1. The net charge on the object is the sum of the charges contributed by each unpaired electron or proton. It is usually measured not in electrons or protons but in coulombs. One coulomb is equal to 6.25 x 1018 unpaired electrons (or protons).
• Coulomb’s Law: Two nonmoving objects in a vacuum with charges Q1 and Q2 respectively will be attracted to each other (if Q1 and Q2 have opposite signs) or repelled (if Q1 and Q2 have the same sign) with a force F (in Newtons) equal to: where r is the distance between the two objects in meters, and ε0 is known as the permittivity constant (= 8.85 x 10-12 coulombs2/newtons•meters2). Note that the amplitude of this force decreases with the square of the distance. It thus can become quite weak at large distances from the object.
• Electric Field: A small test charge moved into any location near enough to a charged object will experience a net electrostatic force. A map of the direction and amplitude of that force at all locations around a charged object defines the electric field around the object. For a single charged object (monopole), the electric field lines of force radiate away from (or towards) the object equally in all directions. If multiple charged objects are present, the amplitude and direction of the net force at any point in the electric field is the vector sum of the component forces present at that point. For two charges of opposite polarity (sign) sufficiently close together (a dipole), the lines of force in the surrounding electric filed are curved.
• Multipolar Fields: The electrical fields generated by biological sources are rarely dipolar and practically never monopolar. Instead, complex arrays of charges will generate many axes around which the charges are distributed. The resulting electrical field is the sum of the effects of the multiple axes. The most important axis is usually the dipole component, the next most important axis the quadrupole component, a third axis the octupole component, etc. The relative contributions of each axis to the overall electrical field depends on the distance between the sampling point and the object. Dipole contributions fall off with distance as 1/r3, quadrupoles as 1/r4, octupoles as 1/r5, etc. As a result of the faster fading of higher order axis contributions, only the dipole component will be detectable at large distances; however, at close distances, all components can contribute significantly to the electric field.
• Electric Potential: The electrostatic potential at any location surrounding a charged object is the amount of work that is required to bring a small unit of positive charge from infinity (where the electric field force surrounding the object is zero) to the location. If the object has an overall positive charge, one must do work against the field’s repellent force to bring the unit charge to the location. The electrostatic potential in this case is said to be positive. If the object has a negative charge, it will exert an attractive force on the positive test charge and thus the work done to bring the test charge closer will be negative. In this case, the electrostatic potential at the final resting location of the test charge is said to be negative. Electric potential is measured in volts and is thus often called the “voltage” at a location. The electric field is the spatial gradient in voltage at any location. The potential difference between any two points in the electric field is simply the difference between the voltages at the two points and is often called the “voltage drop” between the two points. One can connect all points around an object that have the same voltage as isopotential lines. These are equivalent to the lines connecting all locations at the same altitude in a topographic map. Examples of the electric field lines (dashed) and isopotential lines (solid) for a monopole and a dipole are shown below: • Dipole Voltage: The voltage V at a distance r from a dipole and an angle θ (relative to the line joining the two point charges in the dipole) is where Q is the magnitude of the charge on each part of the dipole and δ depends on the distance between them. Thus along the line perpendicular to that joining the two charges and midway between them, the angle θ is 90° and the cos(90°) = 0; thus the voltage along that line is zero. Points outside the dipole but along the line joining the two charges will show the maximal voltage values.

## Electric fields in different media

• General Formulation: The formula given above for the electrical force at any location surrounding two charges Q1 and Q2 in a vacuum can be generalized for two charges in any medium as follows: where all terms are as before and k is the dielectric constant of the medium. A vacuum is a perfect insulator in that the electric force created by the two charges cannot induce any repositioning or movement of other electrons, atoms, or molecules. The dielectric constant for a vacuum is 1.
• Conductors: At the other extreme, are conducting media in which electrons, atoms, and/or molecules are present and free to move under the influence of the electric field. Moveable electrons, for example, in the presence of an electric field will move towards the positive pole leaving their formerly paired positive charges to accumulate near the negative charge. This redistribution of elements of the medium so that opposite charges accumulate around the initial charges continues until it cancels out the electric field inside the conductor. The dielectric constant for a conductor is thus set at infinity, and plugging this value into the above equation, one can see that inside the conductor, the force at any location is zero. By the same token, it will take no work to move a test charge around inside the conductor and thus the voltage inside a conductor is the same everywhere.
• Dielectrics: These are media in which movements of electrons, atoms, and molecules are constrained. However, it is still possible for electrons to move within a medium atom or molecule, or it is possible for a molecule to rotate so that its most positive side faces the negative charge and its negative side faces the positive charge. The parallel alignments of medium molecules or electrons inside an atom or molecule create thousands of tiny dipoles with lines of force opposite to those surrounding the original charges. The result is a reduction in the amplitude of the electrical field surrounding the charges: greater polarization of the medium results in greater diminution of the electrical field. Higher values of the dielectric constant reflect greater susceptibility to polarization and thus a greater reduction in the electrical field inside the medium. The dielectric constant for air is 1.00054, glass 4.7, and freshwater at room temperature about 80. Note that the dielectric constant also affects the measurable voltage at any point inside the medium. For example, the electrical potential surrounding a dipole in a non-conducting but dielectric medium is: ## Electric currents

• Ohm’s Law: Suppose we place an electric dipole in a medium which is a worse conductor than a metal, but a better conductor than most dielectrics. Water is such an example. Water invariably has dissolved materials within it, and many of these, such as salts, break up in water into their component charged ions. The presence of an electric field in water will cause positive and negative ions to move in opposite directions. The ionic trajectories follow the electric field lines. This movement of ions in water (or of electrons in a metal) is called an electric current. The magnitude of an electric current between two points (measured in coulombs/second or amperes) is proportional to the voltage difference between the points. The constant of proportionality between an applied voltage and a resulting current is called the conductance of the medium through which the current is flowing. More often, we use the reciprocal of conductance which is called the resistance. If V is the voltage difference between two points and R is the resistance (in ohms), then the current I (in amperes) depends on these variables according to Ohm’s Law: The convention in physics is that current flows from a region of positive voltage to one of more negative voltage. Note that this is opposite to the actual flow of electrons in a metal (from a negative to positive potential location).
• Resistivity: Resistance in a particular context will be higher the greater the distance that the current must flow, the smaller the cross sectional area through which the current passes, and the worse the material as a conductor. The latter term is characterized by the material’s intrinsic resistivity. Because of the resistance of the water in which we have placed our dipole, there will be a steady current of ions towards that part of the dipole of opposite charge to each ion. If there were no resistance, the initial current would quickly cancel the charge at each end of the dipole due to accumulations of oppositely charged ions. If the resistance is high enough, it may take some time before the dipole is fully neutralized. Alternatively, something may occur near the dipole to restore its charge. In either case, if the electric field is maintained or restored for a sufficiently long period, we can measure the electric potential at various points around the dipole and the amount of current at each location. For a stable source of current in a conducting medium, the potential at location (r,θ) from the dipole is where the medium resistivity, ρ0 and the current I have replaced the permittivity, , and the charge, Q, used for non-conducting media.

• Varying Electric Fields and Impedance: Water and many other materials are both conductors and dielectrics: some current will flow through them, but the resistance is high enough that electric fields are sustained and their ability to be polarized and act as a dielectric permits some build-up of counter-fields within the medium. For static electric fields, this may not be significant. If however, the electric field is changing in magnitude or direction, then the dielectric properties of the medium can become important. In a steady electric field, an electron in a conductor may move the entire length of the conductor. This is called a direct current (DC). Now suppose we apply a sinusoidally varying electric field to the conductor. Electrons will first move one direction and then back the other. This is an alternating (AC) current. The higher the frequency of the alternating field, the less distance any one electron can travel before it has to turn around and go the other way. In a non-conducting dielectric, electrons or polar molecules can move a bit, but they can never move far enough to sustain a steady DC current. However, if an alternating field is applied across such a material, the distance electrons have to travel per cycle may be within the polarizing limitations of the material: the higher the dielectric constant for the material, the slower the frequency of alternation which the material can track and thus carry current. The effective resistances of dielectrics may thus drop if the applied electric field is a varying one. To keep this notion of resistance distinct from classical DC resistivity, the term applied to such dielectrics is capacitative reactance. Capacitative reactance decreases with the dielectric constant of the material and with the frequency of the electric field oscillation. Like resistance, it is measured in ohms. Remember that even if the waveform of the electric field variations is not sinusoidal, it can be considered as the sum of a number of different sinusoids (see Web Topic 2.4). Applying such a non-sinusoidal signal to a dielectric, we will find that the dielectric will act like a high-pass filter since it can more easily track the higher frequency components than the lower frequency ones. The overall impedance of a medium like water to a varying electrical field will thus depend on both the resistivity of the water and on the capacitative reactance of the water at the various frequencies making up the waveform of the changing field.
• Electrical Field Distortion: We have assumed so far that media are unbounded and homogeneous. The resulting electric fields can be called free fields (by analogy with sound). However, most media have boundaries and contain objects whose dielectric and/or resistive properties differ from those of the medium. The usual situation is thus not a free field. Boundaries and objects in the medium will distort and change the field shape from free field conditions. For example, suppose we place a monopole in a medium such as water (left figure below) and then place an objects with a resistivity less than water near to the charge (right): Objects which have lower resistivities than the medium bend the electric field lines in the region between themselves and the charge closer together and towards the object. This region of enhanced electric field magnitude corresponds to a region of very closely spaced isopotential lines and thus a steep gradient in voltage.
Objects which have higher resistivities than the medium (below right) bend the electric field lines away from themselves, lowering the field magnitude in the region between themselves and the charge, and show a flattening of the potential gradient in this region: When many objects of differing conductivities are present, the shape of the field can become highly complex. Boundaries are also significant. If we place our charge near to the air-water interface, or near a non-conducting bottom, the electric field magnitude and potential near to the charge will be twice as great as that for a charge suspended in an unbounded volume of water. This is because current can radiate in all directions in the unbounded case, but can only radiate away from the water’s surface in the bounded example.

See Chapter 7 for examples of how differential resistance and capacitance in nearby objects can be used by some electric fish to discriminate between prey and inedible items, and to navigate through familiar locations.

Go