<< Chapter < Page Chapter >> Page >

Calculating resistor diameter: a headlight filament

A car headlight filament is made of tungsten and has a cold resistance of 0 . 350 Ω size 12{0 "." "350" %OMEGA } {} . If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?

Strategy

We can rearrange the equation R = ρL A size 12{R = { {ρL} over {A} } } {} to find the cross-sectional area A size 12{A} {} of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.

Solution

The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in R = ρL A size 12{R = { {ρL} over {A} } } {} , is

A = ρL R . size 12{A = { {ρL} over {R} } "."} {}

Substituting the given values, and taking ρ size 12{ρ} {} from [link] , yields

A = ( 5.6 × 10 –8 Ω m ) ( 4.00 × 10 –2 m ) 0.350 Ω = 6.40 × 10 –9 m 2 .

The area of a circle is related to its diameter D size 12{D} {} by

A = πD 2 4 . size 12{A = { {πD rSup { size 8{2} } } over {4} } "."} {}

Solving for the diameter D size 12{D} {} , and substituting the value found for A size 12{A} {} , gives

D = 2 A p 1 2 = 2 6.40 × 10 –9 m 2 3.14 1 2 = 9.0 × 10 –5 m . alignl { stack { size 12{D =" 2" left ( { {A} over {p} } right ) rSup { size 8{ { {1} over {2} } } } =" 2" left ( { {6 "." "40"´"10" rSup { size 8{ +- 9} } " m" rSup { size 8{2} } } over {3 "." "14"} } right ) rSup { size 8{ { {1} over {2} } } } } {} #=" 9" "." 0´"10" rSup { size 8{ +- 5} } " m" "." {} } } {}

Discussion

The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because ρ size 12{ρ} {} is known to only two digits.

Got questions? Get instant answers now!

Temperature variation of resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See [link] .) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about 100º C size 12{"100"°C} {} or less), resistivity ρ size 12{ρ} {} varies with temperature change Δ T size 12{DT} {} as expressed in the following equation

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 size 12{α} {} is the temperature coefficient of resistivity    . (See the values of α size 12{α} {} in [link] below.) For larger temperature changes, α size 12{α} {} may vary or a nonlinear equation may be needed to find ρ size 12{ρ} {} . Note that α size 12{α} {} is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has α size 12{α} {} close to zero (to three digits on the scale in [link] ), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.
The resistance of a sample of mercury is zero at very low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.
Tempature coefficients of resistivity α size 12{α} {}
Material Coefficient α (1/°C) Values at 20°C.
Conductors
Silver 3 . 8 × 10 3 size 12{3 "." 8 times "10" rSup { size 8{ - 3} } } {}
Copper 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Gold 3 . 4 × 10 3 size 12{3 "." 4 times "10" rSup { size 8{ - 3} } } {}
Aluminum 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Tungsten 4 . 5 × 10 3 size 12{4 "." 5 times "10" rSup { size 8{ - 3} } } {}
Iron 5 . 0 × 10 3 size 12{5 "." 0 times "10" rSup { size 8{ - 3} } } {}
Platinum 3 . 93 × 10 3 size 12{3 "." "93" times "10" rSup { size 8{ - 3} } } {}
Lead 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Manganin (Cu, Mn, Ni alloy) 0 . 000 × 10 3 size 12{0 "." "000" times "10" rSup { size 8{ - 3} } } {}
Constantan (Cu, Ni alloy) 0 . 002 × 10 3 size 12{0 "." "002" times "10" rSup { size 8{ - 3} } } {}
Mercury 0 . 89 × 10 3 size 12{0 "." "89" times "10" rSup { size 8{ - 3} } } {}
Nichrome (Ni, Fe, Cr alloy) 0 . 4 × 10 3 size 12{0 "." 4 times "10" rSup { size 8{ - 3} } } {}
Semiconductors
Carbon (pure) 0 . 5 × 10 3 size 12{ - 0 "." 5 times "10" rSup { size 8{ - 3} } } {}
Germanium (pure) 50 × 10 3 size 12{ - "50" times "10" rSup { size 8{ - 3} } } {}
Silicon (pure) 70 × 10 3 size 12{ - "70" times "10" rSup { size 8{ - 3} } } {}

Note also that α size 12{α} {} is negative for the semiconductors listed in [link] , meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing ρ size 12{ρ} {} with temperature is also related to the type and amount of impurities present in the semiconductors.

Questions & Answers

what is physics
Rhema Reply
a15kg powerexerted by the foresafter 3second
Firdos Reply
what is displacement
Xolani Reply
movement in a direction
Jason
hello
Hosea
Explain why magnetic damping might not be effective on an object made of several thin conducting layers separated by insulation? can someone please explain this i need it for my final exam
anas Reply
Hi
saeid
hi
Yimam
What is thê principle behind movement of thê taps control
Oluwakayode Reply
while
Hosea
what is atomic mass
thomas Reply
this is the mass of an atom of an element in ratio with the mass of carbon-atom
Chukwuka
show me how to get the accuracies of the values of the resistors for the two circuits i.e for series and parallel sides
Jesuovie Reply
Explain why it is difficult to have an ideal machine in real life situations.
Isaac Reply
tell me
Promise
what's the s . i unit for couple?
Promise
its s.i unit is Nm
Covenant
Force×perpendicular distance N×m=Nm
Oluwakayode
İt iş diffucult to have idêal machine because of FRİCTİON definitely reduce thê efficiency
Oluwakayode
if the classica theory of specific heat is valid,what would be the thermal energy of one kmol of copper at the debye temperature (for copper is 340k)
Zaharadeen Reply
can i get all formulas of physics
BPH Reply
yes
haider
what affects fluid
Doreen Reply
pressure
Oluwakayode
Dimension for force MLT-2
Promise Reply
what is the dimensions of Force?
Osueke Reply
how do you calculate the 5% uncertainty of 4cm?
melia Reply
4cm/100×5= 0.2cm
haider
how do you calculate the 5% absolute uncertainty of a 200g mass?
melia Reply
= 200g±(5%)10g
haider
use the 10g as the uncertainty?
melia
which topic u discussing about?
haider
topic of question?
haider
the relationship between the applied force and the deflection
melia
sorry wrong question i meant the 5% uncertainty of 4cm?
melia
its 0.2 cm or 2mm
haider
thank you
melia
Hello group...
Chioma
hi
haider
well hello there
sean
hi
Noks
hii
Chibueze
10g
Olokuntoye
0.2m
Olokuntoye
hi guys
thomas
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask