Vector fields in cylindrical and spherical coordinates: Difference between revisions

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{{For|other meanings of density|Density (disambiguation)}}
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'''Linear density''' is the measure of a quantity of any characteristic value per unit of length. Linear mass density (the amount of mass per unit length) and linear charge density (the amount of electric charge per unit length) are two common examples used in science and engineering.
 
The term linear density is most often used when describing the characteristics of one-dimensional objects, although linear density can also be used to describe the density of a three-dimensional quantity along one particular dimension.  Just as density is most often used to mean mass density, the term linear density likewise often refers to linear mass density. However, this is only one example of a linear density, as any quantity can be measured in terms of its value along one dimension.
 
==Linear mass density==
Consider a long, thin rod of mass <math>M</math> and length <math>L</math>. To calculate the average linear mass density, <math>\bar\lambda_m</math>, of this one dimensional object, we can simply divide the total mass, <math>M</math>, by the total length, <math>L</math>:
:<math>\bar\lambda_m = \frac{M}{L}</math>
If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, <math>l</math>), we can write:
:<math>m = m(l)</math>
Each infinitesimal unit of mass, <math>dm</math>, is equal to the product of its linear mass density, <math>\lambda_m</math>, and the infinitesimal unit of length, <math>dl</math>:
:<math>dm = \lambda_m dl</math>
The linear mass density can then be understood as the derivative of the mass function with respect to the one dimension of the rod (the position along its length, <math>l</math>).
:<math>\lambda_m = \frac{dm}{dl}</math>
 
The [[SI]] unit of linear mass density is the [[kilogram]] per [[meter]] (kg/m).
 
==Linear charge density==
Consider a long, thin wire of charge <math>Q</math> and length <math>L</math>.  To calculate the average linear charge density, <math>\bar\lambda_q</math>, of this one dimensional object, we can simply divide the total charge, <math>Q</math>, by the total length, <math>L</math>:
:<math>\bar\lambda_q = \frac{Q}{L}</math>
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the rod, <math>l</math>), we can write:
:<math>q = q(l)</math>
Each infinitesimal unit of charge, <math>dq</math>, is equal to the product of its linear charge density, <math>\lambda_q</math>, and the infinitesimal unit of length, <math>dl</math>:<ref>{{Citation
| last1 = Griffiths | first1 = David J.
| title = Introduction to Electrodynamics (2nd Edition)
| place = New Jersey
| publisher = [[Prentice Hall]]
| pages = 64
  | year = 1989
| isbn = 0-13-481367-7 }}</ref>
:<math>dq = \lambda_q dl</math>
The linear charge density can then be understood as the derivative of the charge function with respect to the one dimension of the wire (the position along its length, <math>l</math>)
:<math>\lambda_q = \frac{dq}{dl}</math>
 
The [[SI]] unit of linear charge density is the [[Coulomb]] per [[meter]] (C/m).
 
==Other applications==
In drawing or printing, the term linear density also refers to how densely or heavily a line is drawn.
 
==Units==
{{See also|Units of textile measurement}}
Common units include:
*kilogram per meter
*ounce (mass) per foot
*ounce (mass) per inch
*pound (mass) per yard: used in the North American railway industry for the linear density of [[Rail tracks|rail]]s
*pound (mass) per foot
*pound (mass) per inch
*[[Units of textile measurement#Tex|tex]], a unit of measure for the linear density of [[fiber]]s, defined as the mass in grams per 1,000 meters
*[[Units of textile measurement#Denier|denier]], a unit of measure for the linear density of [[fiber]]s, defined as the mass in grams per 9,000 meters
*[[Units of textile measurement#Tex|decitex]] (dtex), the [[International System of Units|SI]] unit for the linear density of [[fiber]]s, defined as the mass in grams per 10,000 meters
 
== See also ==
* [[Columnar density]]
 
== References==
{{Reflist}}
 
{{DEFAULTSORT:Linear Density}}
[[Category:Density]]

Latest revision as of 19:42, 28 August 2014

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