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Application of Kirchhoff s and Ohm s laws to elementary resistive circuits
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Fundamentals of Electric Circuits
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Power computation for a circuit element Use of the passive sign convention in determining voltage and current directions Solution of simple voltage and current divider circuits Assigning node voltages and mesh currents in an electrical circuit Writing the circuit equations for a linear resistive circuit by applying Kirchhoff s voltage law and Kirchhoff s current law
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CHARGE, CURRENT, AND KIRCHHOFF S CURRENT LAW
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The earliest accounts of electricity date from about 2,500 years ago, when it was discovered that static charge on a piece of amber was capable of attracting very light objects, such as feathers The word itself electricity originated about 600 BC; it comes from elektron, which was the ancient Greek word for amber The true nature of electricity was not understood until much later, however Following the work of Alessandro Volta1 and his invention of the copper-zinc battery, it was determined that static electricity and the current that ows in metal wires connected to a battery are due to the same fundamental mechanism: the atomic structure of matter, consisting of a nucleus neutrons and protons surrounded by electrons The fundamental electric quantity is charge, and the smallest amount of charge that exists is the charge carried by an electron, equal to qe = 1602 10 19 C
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Charles Coulomb (1736 1806) Photo courtesy of French Embassy, Washington, DC
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As you can see, the amount of charge associated with an electron is rather small This, of course, has to do with the size of the unit we use to measure charge, the coulomb (C), named after Charles Coulomb2 However, the de nition of the coulomb leads to an appropriate unit when we de ne electric current, since current consists of the ow of very large numbers of charge particles The other charge-carrying particle in an atom, the proton, is assigned a positive sign, and the same magnitude The charge of a proton is qp = +1602 10 19 C (22)
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Current i = dq/dt is generated by the flow of charge through the cross-sectional area A in a conductor i
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Electrons and protons are often referred to as elementary charges Electric current is de ned as the time rate of change of charge passing through a predetermined area Typically, this area is the cross-sectional area of a metal wire; however, there are a number of cases we shall explore later in this book where the current-carrying material is not a conducting wire Figure 21 depicts a macroscopic view of the ow of charge in a wire, where we imagine q units of charge owing through the cross-sectional area A in t units of time The resulting current, i, is then given by i= q t C s (23)
Figure 21 Current ow in an electric conductor
1 See 2 See
brief biography on page 9 brief biography on page 9
Part I
Circuits
If we consider the effect of the enormous number of elementary charges actually owing, we can write this relationship in differential form: dq C i= (24) dt s The units of current are called amperes (A), where 1 ampere = 1 coulomb/second The name of the unit is a tribute to the French scientist Andr Marie Amp` re3 e e The electrical engineering convention states that the positive direction of current ow is that of positive charges In metallic conductors, however, current is carried by negative charges; these charges are the free electrons in the conduction band, which are only weakly attracted to the atomic structure in metallic elements and are therefore easily displaced in the presence of electric elds
EXAMPLE 21 Charge and Current in a Conductor
Problem
Find the total charge in a cylindrical conductor (solid wire) and compute the current owing in the wire
Solution
Known Quantities: Conductor geometry, charge density, charge carrier velocity Find: Total charge of carriers, Q; current in the wire, I Schematics, Diagrams, Circuits, and Given Data: Conductor length: L = 1 m
Conductor diameter: 2r = 2 10 3 m Charge density: n = 1029 carriers/m3 Charge of one electron: qe = 1602 10 19 Charge carrier velocity: u = 199 10 6 m/s
Assumptions: None Analysis: To compute the total charge in the conductor, we rst determine the volume of
the conductor: Volume = Length Cross-sectional area V = L r 2 = (1 m) 2 10 3 2
m2 = 10 6
Next, we compute the number of carriers (electrons) in the conductor and the total charge: Number of carriers = Volume Carrier density carriers m3 Charge = number of carriers charge/carrier N = V n = 10 6 m3 1029 Q = N qe = 1023 carriers 1602 10 19
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