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Potential Difference (V or E)
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Electrical charges exert an electrostatic force on other charges: like charges repel, opposite charges attract. The force decreases as the distance between two charges increases. Work is done when two charges that initially are separated are brought together. Negative work is done if their polarities are opposite and positive work if they are the same. The greater the values of the charges and the greater their initial separation, the greater the work done. (Work =
where f is electrostatic force and r 1 is the initial distance between the two charges.)
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Potential difference is a measure of this work: The potential difference between two points is the work that must be done to move a unit of positive charge (one coulomb) from one point to the other (ie, it is the potential energy of the charge). One volt (V) is the energy required to move one coulomb a distance of one meter against a force of one newton.
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A potential difference exists within a system whenever positive and negative charges are separated. Charge separation may be generated by a chemical reaction (as in a battery) or by diffusion of two electrolyte solutions with different ion concentrations across a selectively permeable barrier, such as a cell membrane. If a charge separation exists within a conducting medium, charges move between the areas of potential difference: Positive charges are attracted to the region with a more negative potential, and negative charges to the region of positive potential.
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Current is defined as the net movement of charge per unit time. According to convention, the direction of current is defined as the direction of flow of positive charge. In metallic conductors current is carried by negatively charged electrons, which move in the opposite direction of conventionally defined current. In nerve and muscle cells current is carried by both positive and negative ions in solution. One ampere (A) of current represents the movement of one coulomb (of charge) per second.
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Any object through which electrical charges can flow is called a conductor. The unit of electrical conductance is the siemens (S). According to Ohm's law the current that flows through a conductor is directly proportional to the potential difference across it:1
I = V×g
Current (A) = Potential difference (V)
×Conductance (S).
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As charge carriers move through a conductor, some of their electrical potential energy is converted into thermal energy caused by their frictional interactions with the conducting medium.
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Each type of material has an intrinsic property called conductivity (σ), which is determined by its molecular structure. Metallic conductors conduct electricity extremely well and thus have high conductivities. Aqueous solutions with high-ionized salt concentrations have somewhat lower conductivity, and lipids have very low conductivity—they are poor conductors of electricity and are therefore good insulators. The conductance of an object is proportional to σ times its cross-sectional area divided by its length:
g = (σ) × area/length.
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Length is defined as the direction along which one measures conductance. For example, the conductance measured along the cytoplasmic core of an axon is reduced if its length is increased or its diameter decreased (Figure A–1).
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Electrical resistance (R) is the reciprocal of conductance and a measure of the resistance provided by an object to current. Resistance is measured in ohms (Ω):
1 Ω = 1/(1 S).
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A capacitor consists of two conducting plates separated by an insulating layer. Its fundamental property is its ability to separate charges of opposite sign: Positive charges are stored on one plate, negative charges on the other. In the example in Figure A–2 a net excess of positive charges on plate x and an equal excess of negative charges on plate y results in a potential difference between the two plates.
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This potential difference can be measured by determining how much work is required to move a positive test charge from y to x. Initially, the test charge is attracted by the negative charges on y and repelled by the more distant positive charges on x. The result of these electrostatic interactions is a force f that opposes the movement of the charge from y to x. However, as the test charge is moved toward x, the attraction by the negative charges on y diminishes and the repulsion by the positive charges on x increases, with the result that the net electrostatic force exerted on the test charge is constant everywhere between x and y. Work (W) is force times the distance (D) over which the force is exerted:
W = f × D.
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The work done in moving the test charge from one side of the capacitor to the other is equal to the difference in electrical potential energy, or potential difference, between x and y. In Figure A–2 it is shown as the shaded region in the plots.
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Capacitance is measured in farads (F). The greater the density of charges on the capacitor plates, the greater the force acting on the test charge and the greater the resulting potential difference across the capacitor (see Figure A–2B). Thus, for a given capacitor there is a linear relationship between the amount of charge (Q) stored on its plates and the potential difference across it:
Q (coulombs) = C (farads) × V (volts) (A–1)
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where C, the capacitance, is a constant.
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The capacitance of a parallel-plate capacitor is determined by two features of its geometry: the area (A) of the two plates and the distance (D) between them. Increasing the area of the plates increases capacitance because a greater amount of charge must be deposited on each side to produce the same charge density, which is what determines the force f acting on the test charge (Figure A–2A and C). Increasing the distance between the plates does not change the force acting on the test charge, but it does increase the work that must be done to move it from one side of the capacitor to the other (Figure A–2A and D). Therefore, for a given charge separation between the two plates, the potential difference between them is proportional to the distance. Put another way, the greater the distance, the smaller the amount of charge that must be deposited on the plates to produce a given potential difference, and therefore the smaller the capacitance (Equation A–1).
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These geometrical determinants of capacitance can be summarized by the equation
C ∞ A/D.
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As shown in Equation A–1, the separation of positive and negative charges on the two plates of a capacitor results in a potential difference between them. The converse of this statement is also true: The potential difference across a capacitor is determined by the net positive and negative charge on its plates. For the potential across a capacitor to change, the amount of electrical charges stored on the two conducting plates must change first.