What is Dimensional Analysis?

Dimensional Analysis is a technique for studying dimensions of physical quantities. It can also be used for other purposes as well, some of these other purposes are as following: –

  • Reducing physical properties of derived SI units into those of the more fundamental SI base units
  • Assist in converting quantities expressed in non-SI units to SI units
  • Verfiying correctness of an equation in terms of dimensional and unitary consistency
  • Determining dimension and unit of a variable in an equation

A Dimension Symbol is represented as a capital letter enclosed in square brackets. For example – The dimension for time is [T], for mass it’s [M]. Below is a table containing Dimensional Symbols of SI Base units.

Dimension Symbols of SI Base Physical Quantities

Physical QuantityDimension Symbol
Time[T]
Length[L]
Mass[M]
Luminous Intensity[J]
Thermodynamic Temperature[\Theta]
Electric Current[I]
Amount of Substance[N]

Dimensional Equations

For an equation describing some physical situation to be true, it need to have same dimensions on both sides of the equation. For instance – The velocity equation v=d / t to be true on both sides of the equation v and d / t need to have same dimensions.

Dimensional Symbols for Derived SI Units

Physical QuantitySI UnitDimensional Symbol
Plane Angleradian[L] .[L]^{-1}
Solid Anglesteradian[L]^{2} \cdot[L]^{-2}
Frequencyhertz[T]^{-1}
Forcenewton[L] \cdot[M] \cdot[T]^{-2}
Pressurepascal[L]^{-1} \cdot[M] \cdot[T]^{-2}
Energyjoule[L]^{2} \cdot[M] \cdot[T]^{-2}
Powerwatt[L]^{2} \cdot[M] \cdot[T]^{-3}
Electric Chargecoulomb[T] \cdot[I]
Potential Differencevolt[L]^{2} \cdot[M] \cdot[T]^{-3} \cdot[I]^{-1}
Capacitancefarad[L]^{-2} \cdot[M]^{-1} \cdot[T]^{4} \cdot[I]^{2}
Electrical Resistanceohm[L]^{2} \cdot[M] \cdot[T]^{-3} \cdot[I]^{-2}
Magnetic Fluxweber[L]^{2} \cdot[M] \cdot[T]^{-2} \cdot[I]^{-1}
Magnetic Flux Denstiytesla[M] .[T]^{-2} \cdot[I]^{-1}
Inductancehenry[L]^{2} \cdot[M] \cdot[T]^{-2} \cdot[I]^{-2}
Luminous Fluxlumen[J]
illuminancelux[L]^{-2} \cdot[J]
Activitybequerel[T]^{-1}
Areasquare metre[L]^{2}
Volumecubic metre[L]^{3}
Speed/Velocitymetre per second[L] .[T]^{-1}
Accelerationmetre per socond squared[L] \cdot[T]^{-2}
Wavenumberreciprocal metre[L]^{-1}
Density, Mass Densitykilogram per cubic metre[M] \cdot[L]^{-3}
Specific Volumecubic metre per kilogram[L]^{3} \cdot[M]^{-1}
Current Densityampere per square metre[I] \cdot[L]^{-2}
Magnetic Field Strengthampere per metre[I] \cdot[L]^{-1}
Concentration of amount of substancemole per cubic metre[N] .[L]^{-3}
Luminancecandela per square metre[J] .[L]^{-2}
Rate of Coolingkelvin per second[\Theta] .[T]^{-1}

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