Circuit Elements

The amount of charge moving through a conductor per second is known as current. The unit of current is the coulomb per second, or ampere.2 The symbol I is used to represent current. Even though it is electrons that are considered to be the carriers of charge, the convention for a long time has been to take the direction of the current as if it were positive charges that flow. This is, therefore, known as conventional current and is generally used to describe current flow. Current does not flow unimpeded by conductors. The opposition to the flow of charges offered by a material is known as its resistance.2 The unit of resistance is the ohm (Ω). It may help to think of resistance as friction through conductors. Resistance depends on several factors, including the type of material, length, temperature, and cross-sectional area. Resistance is lower in conductors and higher in insulators. Resistance is directly proportional to length, meaning that resistance increases as length increases. Generally, resistance increases with temperature. It is inversely proportional to the cross-sectional area, which means that resistance increases as a wire becomes thinner. The relationship among current, resistance, and voltage is described by Ohm’s Law, which states that resistance is directly proportional to voltage and indirectly proportional to current

ohms-law

A parallel circuit is one in which two or more resistors are arranged in such a way that current has more than one path through which to flow and each resistor has the same potential difference across it.2

parallel-circuit

The total resistance is the inverse sum of the inverse resistances.

parallel-resistance

The total current is the sum of the currents through each resistor.

IT = I1 + I2 + I3

The voltage across each resistance is the same.

VT = V1 = V2 = V3

Some materials are intrinsically better conductors of electricity than others. For example, copper conducts electricity better than plastic, which is why electrical wires have a copper core surrounded by a layer of plastic rather than the other way around. The number that characterizes the intrinsic resistance to current flow in a material is called the resistivity (ρ), for which the SI unit is the ohm–meter (Ω · m).2

If two conducting plates are connected to a battery, charge flows from the battery to the plates. One plate becomes positively charged and the other becomes negatively charged. The plates form a capacitor, which stores charge. A capacitor consists of two conductors separated by an insulator. That insulator could be air, oil, paper, or other materials known as dielectrics. The capacitance, C, is the charge per unit voltage of the plates and is represented by the following equation, in which “q” is the charge of one of the plates and “V” is the voltage across the plates.

Capacitance

When Q is measured in Coulombs and V is measured in volts, the unit of capacitance is the farad, F. One farad is a large unit of capacitance. The charge on a capacitor does not change instantaneously. The change in charge occurs over time interval Δt.

change-in-charge

Charge flowing into a capacitor builds up rather than passing through it. That charge cannot build up indefinitely. Instead, the charge builds up until the voltage balances the external voltage pushing charge on the capacitor. The capacitance of a parallel plate capacitor is dependent upon the geometry of the two conduction surfaces. For the simple case of the parallel plate capacitor, the capacitance is given by:

parallel-plate-capacitance

where ε0 is the permittivity of free space, 8.85 x 10-12 F/m, A is the area of overlap of the two plates, and d is the separation of the two plates. The separation of charges sets up an electric field between the plates of a parallel plate capacitor. The electric field between the plates of a parallel plate capacitor is a uniform electric field with parallel field vectors, the magnitude of which can be calculated as:

electric-field

electrostatics equations. If

E = (Qk)/r2        and

V= (kQ)/r,    then

V = E × rr

in this setup is the distance between the plates, d, so we can rewrite this as:

V = Ed.

The direction of the electric field at any point between the plates is from the positive plate toward the negative plate. If we imagine placing a positively charged particle between the oppositely charged plates, we would expect the particle to accelerate in that same direction. This should not be surprising, as electric field lines always point in the direction a force would be exerted on a positive charge.

Regardless of the particular geometry of a capacitor (parallel plate or otherwise), the function of a capacitor is to store an amount of energy in the form of charge separation at a particular voltage. This is akin to the function of a dam, the purpose of which is to store gravitational potential energy by holding back a mass of water at a given height. The potential energy stored in a capacitor is:

capacitor-potential-energy

The term dielectric material is just another way of saying insulation. When a dielectric material, such as air, glass, plastic, ceramic, or certain metal oxides, is introduced between the plates of a capacitor, it increases the capacitance by a factor called the dielectric constant (κ). The dielectric constant of a material is a measure of its insulating ability, and a vacuum has a dielectric constant of 1, by definition. For reference, the dielectric constant of air is just slightly above 1, glass is 4.7, and rubber is 7.

C′ = κC

where C′ is the new capacitance with the dielectric present and C is the original capacitance.

When a dielectric material is placed in an isolated, charged capacitor, that is, a charged capacitor disconnected from any circuit, the voltage across the capacitor decreases. This is the result of the dielectric material shielding the opposite charges from each other. By lowering the voltage across a charged capacitor, the dielectric has increased the capacitance of the capacitor by a factor of the dielectric constant. Thus, when a dielectric material is introduced into an isolated capacitor, the increase in capacitance arises from a decrease in voltage. When a dielectric material is placed in a charged capacitor within a circuit, that is, still connected to a voltage source, the charge on the capacitor increases. The voltage must remain constant because it must be equal to that of the voltage source. By increasing the amount of charge stored on the capacitor, the dielectric has increased the capacitance of the capacitor by a factor of the dielectric constant. Thus, when a dielectric material is introduced into a circuit capacitor, the increase in capacitance arises from an increase in stored charge. The stored energy in a capacitor is only useful if it is allowed to discharge. The charge can be released from the plates either by discharging across the plates or through some conductive material with which the plates are in contact. For example, capacitors can discharge into wires, causing a current to pass through the wires in much the same way that batteries cause current to move through a circuit. The paddles of the defibrillator machine, once charged, are placed on either side of a patient’s heart that has gone into a life-threatening arrhythmia (such as ventricular fibrillation).

When capacitors are connected in series, the total capacitance decreases in similar fashion to the decreases in resistance seen in parallel resistors

capacitor-series

This is because the capacitors must share the voltage drop in the loop and therefore cannot store as much charge. Functionally, a group of capacitors in series acts like one equivalent capacitor with a much larger distance between its plates (in fact, with a distance equal to those of each of the series capacitors added together). This increase in distance, as seen earlier, means a smaller capacitance.

series-capacitance

which shows that Cs decreases as more capacitors are added. Note that for capacitors in series, the total voltage is the sum of the individual voltages, just like resistors in series. Capacitors wired in parallel produce a resultant capacitance that is equal to the sum of the individual capacitances.

capacitor-parallel 

Therefore, Cp increases as more capacitors are added:

Cp = C1 + C2 + C3 + ⋯ + Cn

Just as we saw with resistors in parallel, the voltage across each parallel capacitor is the same and is equal to the voltage across the entire combination.2

Conductivity can be divided into two categories: metallic conductivity, as seen in solid metals and the molten forms of some salts, or electrolytic conductivity, as seen in solutions. Conductance is the reciprocal of resistance. The SI unit for conductance is the siemens (S), sometimes given as siemens per meter (S//m) for conductivity. Some materials allow free flow of electric charge within them; these materials are called electrical conductors. Metal atoms can easily lose one or more of their outer electrons, which are then free to move around in the larger collection of metal atoms. This makes most metals good electrical and thermal conductors. The metallic bond has often been visualized as a sea of electrons flowing over and past a rigid lattice of metal cations. Metallic bonding is more accurately described as an equal distribution of the charge density of free electrons across all of the neutral atoms within the metallic mass. While not substantially different from metallic conductivity, it is important to note that electrolytic conductivity depends on the strength of a solution. Distilled deionized water has such a low ion concentration that it may be considered an insulator, while sea water and orange juice are excellent conductors. Conductivity in an electrolyte solution is measured by placing the solution as a resistor in a circuit and measuring changes in voltage across the solution. Because concentration and conductivity are directly related, this method is often used to determine ionic concentrations in solutions, such as blood. One caveat is that conductivity in non-ionic solutions is always lower than in ionic solutions. While the concentration of total dissolved solids does relate to conductivity, the contribution of non-ionic solids is much, much less important than ion concentration. Conductivity is affected by electrolyte concentration. No electrolyte will result in no ionization and no conductivity. Too much electrolyte, however, results in ions being too crowded and there is less ion mobility. This in turn causes less conductivity. In metals, conductivity decreases as temperature increases and in semiconductors, conductivity increases as temperature increases. At extremely low temperatures (below a certain critical temperature typically a few degrees above absolute zero), some materials have superconductivity. This is when there is virtually no resistance to current flow, causing the current to loop almost forever under such conditions. Conductivity (σ) is the inverse of resistivity (ρ).

 

References

1) Matthew Sadiku (2009). Elements of electromagnetics. p. 104.

2) Farago, P.S. (1961) An Introduction to Linear Network Analysis, pp. 18–21, The English Universities Press Ltd.

3) Keithley, Joseph F (1999). Daniell Cell. John Wiley and Sons. pp. 49–51.

4) Verschuur, Gerrit L. (1993). Hidden Attraction: The History and Mystery of Magnetism. New York: Oxford University Press. p. 76

5) Dorland’s (2012). Dorland’s Illustrated Medical Dictionary (32nd ed.). Elsevier Saunders.

p.1862.

6) Black, J.A., Sontheimer, H., Oh, Y., and Waxman, S.G. (1995). In The Axon, S. Waxman, J. Kocsis, and P. Stys, eds. Oxford University Press, New York, pp. 116–143.

 

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  • ohms-law
  • parallel-circuit
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  • Capacitance
  • change-in-charge
  • parallel-plate-capacitance
  • electric-field
  • capacitor-potential-energy
  • series-capacitance
  • capacitor-series
  • capacitor-parallel
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