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The resistance of an object also depends on temperature, since R 0 size 12{R rSub { size 8{0} } } {} is directly proportional to ρ size 12{ρ} {} . For a cylinder we know R = ρL / A size 12{R=ρL/A} {} , and so, if L size 12{L} {} and A size 12{A} {} do not change greatly with temperature, R size 12{R} {} will have the same temperature dependence as ρ size 12{ρ} {} . (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on L size 12{L} {} and A size 12{A} {} is about two orders of magnitude less than on ρ size 12{ρ} {} .) Thus,

R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}

is the temperature dependence of the resistance of an object, where R 0 size 12{R rSub { size 8{0} } } {} is the original resistance and R size 12{R} {} is the resistance after a temperature change Δ T size 12{DT} {} . Numerous thermometers are based on the effect of temperature on resistance. (See [link] .) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.

A photograph showing two digital thermometers used for measuring body temperature.
These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)

Calculating resistance: hot-filament resistance

Although caution must be used in applying ρ = ρ 0 ( 1 + α Δ T ) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {} and R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {} for temperature changes greater than 100º C size 12{"100"°"C"} {} , for tungsten the equations work reasonably well for very large temperature changes. What, then, is the resistance of the tungsten filament in the previous example if its temperature is increased from room temperature ( 20ºC ) to a typical operating temperature of 2850º C size 12{"2850"°"C"} {} ?

Strategy

This is a straightforward application of R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {} , since the original resistance of the filament was given to be R 0 = 0 . 350 Ω size 12{R rSub { size 8{0} } =0 "." "350"` %OMEGA } {} , and the temperature change is Δ T = 2830º C size 12{ΔT="2830"°"C"} {} .

Solution

The hot resistance R size 12{R} {} is obtained by entering known values into the above equation:

R = R 0 ( 1 + α Δ T ) = ( 0 . 350 Ω ) [ 1 + ( 4.5 × 10 –3 / ºC ) ( 2830º C ) ] = 4.8 Ω.

Discussion

This value is consistent with the headlight resistance example in Ohm’s Law: Resistance and Simple Circuits .

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Phet explorations: resistance in a wire

Learn about the physics of resistance in a wire. Change its resistivity, length, and area to see how they affect the wire's resistance. The sizes of the symbols in the equation change along with the diagram of a wire.

Resistance in a Wire

Section summary

  • The resistance R size 12{R} {} of a cylinder of length L size 12{L} {} and cross-sectional area A size 12{A} {} is R = ρL A size 12{R = { {ρL} over {A} } } {} , where ρ size 12{ρ} {} is the resistivity of the material.
  • Values of ρ size 12{ρ} {} in [link] show that materials fall into three groups— conductors, semiconductors, and insulators .
  • Temperature affects resistivity; for relatively small temperature changes Δ T size 12{DT} {} , resistivity is ρ = ρ 0 ( 1 + α Δ T ) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {} , where ρ 0 size 12{ρ rSub { size 8{0} } } {} is the original resistivity and α is the temperature coefficient of resistivity.
  • [link] gives values for α size 12{α} {} , the temperature coefficient of resistivity.
  • The resistance R size 12{R} {} of an object also varies with temperature: R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ ΔαT \) } {} , where R 0 size 12{R rSub { size 8{0} } } {} is the original resistance, and R is the resistance after the temperature change.

Conceptual questions

In which of the three semiconducting materials listed in [link] do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)

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Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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