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Current and Resistance

# Current and Resistance - Chapter 6 Current and Resistance...

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Chapter 6 Current and Resistance 6.1 Electric Current ..................................................................................................... 6-2 6.1.1 Current Density .............................................................................................. 6-2 6.2 Ohm’s Law ........................................................................................................... 6-4 6.3 Electrical Energy and Power ................................................................................. 6-7 6.4 Summary ............................................................................................................... 6-8 6.5 Solved Problems ................................................................................................... 6-9 6.5.1 Resistivity of a Cable ..................................................................................... 6-9 6.5.2 Charge at a Junction ..................................................................................... 6-10 6.5.3 Drift Velocity ............................................................................................... 6-11 6.5.4 Resistance of a Truncated Cone ................................................................... 6-12 6.5.5 Resistance of a Hollow Cylinder ................................................................. 6-13 6.6 Conceptual Questions ......................................................................................... 6-14 6.7 Additional Problems ........................................................................................... 6-14 6.7.1 Current and Current Density ........................................................................ 6-14 6.7.2 Power Loss and Ohm’s Law ........................................................................ 6-14 6.7.3 Resistance of a Cone .................................................................................... 6-15 6.7.4 Current Density and Drift Speed .................................................................. 6-15 6.7.5 Current Sheet ............................................................................................... 6-16 6.7.6 Resistance and Resistivity ............................................................................ 6-16 6.7.7 Power, Current, and Voltage ........................................................................ 6-17 6.7.8 Charge Accumulation at the Interface ......................................................... 6-17 6-1

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Current and Resistance 6.1 Electric Current Electric currents are flows of electric charge. Suppose a collection of charges is moving perpendicular to a surface of area A , as shown in Figure 6.1.1. Figure 6.1.1 Charges moving through a cross section. The electric current is defined to be the rate at which charges flow across any cross- sectional area. If an amount of charge Q passes through a surface in a time interval t , then the average current avg I is given by avg Q I t = (6.1.1) The SI unit of current is the ampere (A), with 1 A = 1 coulomb/sec. Common currents range from mega-amperes in lightning to nano-amperes in your nerves. In the limit the instantaneous current I may be defined as 0, t ∆ → dQ I dt = (6.1.2) Since flow has a direction, we have implicitly introduced a convention that the direction of current corresponds to the direction in which positive charges are flowing. The flowing charges inside wires are negatively charged electrons that move in the opposite direction of the current. Electric currents flow in conductors: solids (metals, semiconductors), liquids (electrolytes, ionized) and gases (ionized), but the flow is impeded in non- conductors or insulators. 6.1.1 Current Density To relate current, a macroscopic quantity, to the microscopic motion of the charges, let’s examine a conductor of cross-sectional area A , as shown in Figure 6.1.2. 6-2
Figure 6.1.2 A microscopic picture of current flowing in a conductor. Let the total current through a surface be written as I d = ∫∫ J A G G (6.1.3) where is the current density (the SI unit of current density are ). If q is the charge of each carrier, and n is the number of charge carriers per unit volume, the total amount of charge in this section is then J G 2 A/m ( ) Q q nA x = . Suppose that the charge carriers move with a speed ; then the displacement in a time interval t will be d v d x v t = , which implies avg d Q I nqv A t = = (6.1.4) The speed at which the charge carriers are moving is known as the drift speed . Physically, is the average speed of the charge carriers inside a conductor when an external electric field is applied. Actually an electron inside the conductor does not travel in a straight line; instead, its path is rather erratic, as shown in Figure 6.1.3.

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Current and Resistance - Chapter 6 Current and Resistance...

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