thumb|317x317px|The linear density, represented by λ, indicates the amount of a quantity, indicated by m, per unit length along a single dimension.

Linear mass density or simply linear density is defined in the International System of Quantities (ISQ) as the quotient of mass and length. It is also called titer in textile engineering.

The SI unit of linear mass density is the kilogram per meter (kg/m).

Although (linear) density is most often used to mean (linear) mass density, the concept can be generalized for the quotient of any other quantity by length, called lineic quantities in the ISQ.

Common units include:

  • tex, a unit of measure for the linear density of fibers, defined as the mass in grams per 1,000 meters
  • denier, a unit of measure for the linear density of fibers, defined as the mass in grams per 9,000 meters
  • decitex (dtex), a unit for the linear density of fibers, defined as the mass in grams per 10,000 meters

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Generalization: lineic quantities

The qualifier lineic is recommended in the International System of Quantities (ISO 80000-1) to denote the quotient of any physical quantity by length.

The expressions "per unit length" or "linear ... density" (or simply "density") are also often used, with resulting units involving reciprocal metre (m<sup>−1</sup>), for example:

  • linear mass density or lineic mass
  • linear charge density or lineic electric charge, electric charge per unit length
  • linear number density or lineic number, number of entities per unit length
  • propagation constant (attenuation constant and phase constant)

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 display="block">\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 wire, <math>l</math>), we can write:

<math display="block">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>:

<math display="block">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 display="block">\lambda_q = \frac{dq}{dl}</math>

Notice that these steps were exactly the same ones we took before to find <math display="inline">\lambda_m = \frac{dm}{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.

The most famous abstraction of linear density is the probability density function of a single random variable.

See also

  • Density
  • Area density
  • Columnar density
  • Paper density
  • Linear number density

References