# System of units

A system of units , previously a system of measurement , is a collection of units of measurement in which exactly one unit is assigned to each type of size . In Germany, the International System of Units (SI, for systême international) is generally used. Other systems of units are the CGS system or the Anglo-American measurement system .

## meaning

Physical quantities can only ever be specified as a multiple of a unit of measurement (short: unit). This is the equation for the relationship between place , time and speed with non-accelerated movement

${\ displaystyle {\ frac {x} {x_ {0}}} = K \ cdot {\ frac {v} {v_ {0}}} \ cdot {\ frac {t} {t_ {0}}}}$

where is the unit of length, the unit of speed and the unit of time. is a real constant of proportionality that depends on the choice of units. ${\ displaystyle x_ {0}}$${\ displaystyle v_ {0}}$${\ displaystyle t_ {0}}$${\ displaystyle K}$

The constants can be summarized and obtained by transforming this equation

${\ displaystyle x = C \ cdot v \ cdot t}$

With

${\ displaystyle C = K \ cdot {\ frac {x_ {0}} {v_ {0} \ cdot t_ {0}}}}$.

For example, if the location is given in meters (m), the time in seconds (s) and the speed in multiples of the vacuum speed of light ( ), then is and the constant is ${\ displaystyle c}$${\ displaystyle K = 299 \, 792 \, 458}$${\ displaystyle C}$

${\ displaystyle C = 299 \, 792 \, 458 \ \ mathrm {\ frac {m} {s \ cdot c}}}$

So if you have, for example, a speed of 0.5 c and a time of 2 s, the equation results

${\ displaystyle x = 299 \, 792 \, 458 \ \ mathrm {\ frac {m} {s \ cdot c}} \ cdot 0 {,} 5 \ \ mathrm {c} \ cdot 2 \ \ mathrm {s} = 299 \, 792 \, 458 \ \ mathrm {m}}$

- a conclusive result.

Since it is impractical to include such a constant in every equation, it makes sense to choose units so that many constants become 1. As defined one (thus m / s according to the above example, the unit of velocity in meter / second ), and thus the constant is found to be in the above equation , then what the familiar equation ${\ displaystyle v_ {0} = {\ frac {x_ {0}} {t_ {0}}}}$${\ displaystyle C = 1}$

${\ displaystyle x = v \ cdot t}$

results.

The constant in this equation says something about the system of units used. Many natural constants are in truth "unit system constants". The Boltzmann constant is nothing more than a conversion factor between energy and temperature (which is why the temperature is often given in units of energy). So it actually says nothing about nature , only something about the temperature scale used. ${\ displaystyle k _ {\ mathrm {B}}}$

## variants

While it makes little sense to define a system of units in which it does not apply, for reasons of visualization, different ways of writing size equations have become established especially for the physical quantities of electrodynamics . This is the first Maxwell equation in a vacuum in SI units ${\ displaystyle x = vt}$

${\ displaystyle \ operatorname {div \,} {\ vec {E}} = {\ frac {\ rho} {\ varepsilon _ {0}}},}$

in Gaussian cgs units

${\ displaystyle \ operatorname {div \,} {\ vec {E}} = 4 \ pi \ rho,}$

and in Heaviside-Lorentz units (also called rationalized cgs)

${\ displaystyle \ operatorname {div \,} {\ vec {E}} = \ rho.}$

From the point of view of the SI, these spellings differ only in that the constant is arbitrarily set equal to a number in the two CGS systems . This has the consequence that the electric current strength loses the character of a basic quantity in these unit systems; in addition, units of measurement and dimensions are ambiguous: a size value such as B. 2.0 cm can then be the measure of a length, but also z. B. also that of the capacitance of a capacitor. ${\ displaystyle \ varepsilon _ {0}}$

Some important systems of units are: