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Ch0211 - Chapter 11 Resistivity Power 11 Resistivity Power...

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Chapter 11 Resistivity, Power 84 11 Resistivity, Power In chapter 9 we discussed resistors that conform to Ohm’s Law. From the discussion, one could deduce that the resistance of such a resistor depends on the nature of the material of which the resistor is made and on the size and shape of the resistor. In fact, for resistors made out of a single kind of material, in the shape of a wire 1 with a terminal at each end, the resistance is given by: A L R r = (11-1) where: R is the resistance of the resistor as measured between the ends, r is the resistivity of the substance of which the resistor is made, A is the cross-sectional area of the wire-shaped resistor, and L is the length of the resistor. The values of resistivity for several common materials are provided in the following table: Material Resistivity r Silver 1 . 6 × 10 8 Ω⋅ m Copper 1 . 7 × 10 8 Ω⋅ m Gold 2 . 4 × 10 8 Ω⋅ m Aluminum 3 × 10 8 Ω⋅ m Tungsten 5 . 6 × 10 8 Ω⋅ m Nichrome 1 . 0 × 10 6 Ω⋅ m Seawater 0 . 25 Ω⋅ m Rubber 1 × 10 13 Ω⋅ m Glass 1 × 10 10 to 1 × 10 14 Ω⋅ m Quartz 5 × 10 15 to 7 . 5 × 10 17 Ω⋅ m 1 The resistor can have any shape such that one linear dimension can be identified as the length of the resistor, and, such that the intersection of a plane perpendicular to the length of the resistor, at any position along the length of the resistor, has one and the same area (the cross-sectional area of the resistor). I am calling the shape “the shape of a wire” for ease in identification of what we mean by the “along-the-length” dimension. Cross-sectional Area A Length L
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Chapter 11 Resistivity, Power 85 In the expression A L R r = , the resistivity r depends on the charge carrier 2 density, that is, the number-of-charge-carriers-per-volume. The more charge carriers per volume, the smaller the resistance since, for a given velocity of the charge carriers, more of them will be passing any point along the length of the resistor every second for a given voltage across the resistor. The resistivity also depends on the retarding force factor. We said that the retarding force on each charge carrier is proportional to the velocity of that charge carrier. Retarding Force = (factor) times (charge carrier velocity) (The minus sign is there because the retarding force is in the direction opposite that of the charge-carrier velocity.) The bigger the retarding force factor, the greater the resistivity of the material for which the factor applies. The charge carrier density and the retarding force factor determine the value of r . The effect of r on the resistance is evident in the expression A L R r = . The bigger r is, the greater the resistance is. Why the factor of L in A L R r = ? It’s saying that the greater the length of the single-substance resistor in the shape of a wire, the greater the resistance of the resistor, all other things being equal (same substance, same cross-sectional area). It means, for instance, that if you have two
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