Dimension consideration

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The dimensional observation or dimensional test is a trivial method for checking whether an equation with physical quantities can be correct. The dimensions of the terms on either side of the equation must match. Dimensional correctness is a necessary condition for physical correctness. However, it is not a sufficient condition that the equation as a whole applies in terms of content and numerical values.

regulate

  • An equation can only express a physical relationship if its two sides are of the same dimension.
  • In sums and differences, all terms must be of the same dimension.
  • In products and quotients, terms of different dimensions can be linked to one another.
  • Transcendent functions such as , or are only defined for one argument , which is a quantity of the dimension number . The result also has the dimension number.

Examples

Determine the dimension of a proportionality factor

The equation for the gravitational force of the masses m and M , which are at a distance r , should serve as an example . In simplified form (not vectorial ) it reads

Wanted is the dimension of the gravitational constant G . Solving the equation by G results

If one knows the dimension of all sizes on the right side, the dimension of the left side results with the dimension symbols of the international size system :

The opposite way is also possible: One recognizes a difference between the dimension of the left and right side of the equation, determines the dimension of the obviously missing factor and can then sometimes guess which quantity is still missing.

Buckingham's Π-theorem

Assuming that a proportionality factor Π (large Pi) has the dimension number, Buckingham's Π theorem can be used to derive the relationship between the quantities used in a formula. If, for example, it is known that the mass of a homogeneous sphere only depends on the density and the sphere radius, then the formula can be determined as follows:

where a constant of dimension is number, the mass of the sphere, the radius and the density. The dimension equation with the dimension for mass and for length, i.e. with must:

It follows

  • for the exponents of  :
  • for the exponents of  :  

The solution of these two equations gives: and . That means: and after resolved:

The value of the constant ( ) cannot be determined with this theorem. An approximation could be found empirically by weighing any ball of known density and radius.

Argument of a transcendent function

When a capacitor is discharged through a resistor , the voltage runs as a decaying exponential function , the time is in the exponent:

The dimension of the factor must therefore be that of an inverse time so that the product takes on the dimension number. Since the resistor is also involved in addition to the capacitance of the capacitor , one can already assume that the proportionality factor is mathematically formed from these quantities. The dimension of the product of capacitance and resistance has the dimension of time. It is therefore obvious that the following can be traced back to the given quantities:

See also

literature

  • Hans Dieter Baehr: Physical quantities and their units - an introduction for students, natural scientists and engineers. Volume 19 of the series of study books on natural science and technology , Bertelsmann Universitätsverlag, Düsseldorf 1974. ISBN 3-571-19233-8
  • Hans Rupp: Physical quantities, formulas, laws and definitions. 2nd edition, Oldenbourg Schulbuchverlag, June 1995. ISBN 3-486-87093-9
  • Paul A. Tipler : Physics . 3rd corrected reprint of the 1st edition 1994, Spektrum Akademischer Verlag Heidelberg Berlin, 2000, ISBN 3-86025-122-8