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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.

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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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How we can toraidal magnetic field
Aditya Reply
How we can create polaidal magnetic field
Aditya
4
Mykayuh Reply
Because I'm writing a report and I would like to be really precise for the references
Gre Reply
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!
Gre Reply
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Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
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