The nature of physical quantities is described by its dimensions. All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities. We shall call these base quantities as the seven dimensions of the physical world, which are denoted with square brackets [ ]. Thus, length has the dimension [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd], and amount of substance [mol].
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
- Volume of an object V = l × b × h = (length)3 , thus [V] = [L3] . Volume is independent of mass and time, so we may write [V] = [M0L3T0].
- Velocity (v) = displacement/time. Therefore, [v]=[ M0L1T-1]. The velocity has zero dimension in mass, one dimension in length and -1 dimension in time.
Dimensional Formulae and Dimensional Equations
The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.
Examples: the dimensional formula of the volume is [M0 L3 T0], and that of speed or velocity is [M0 L1 T-1]. Similarly, [M0 L1 T-2] is the dimensional formula of acceleration and [M1 L-3 T0] that of mass density.
An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. The dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities.
Examples: the dimensional equations of volume [V], speed [v], force [F ] and mass density [ñ] may be expressed as
[V] = [M0 L3 T0]
[v] = [M0 L T-1]
[F] = [M L T-2]
[ρ] = [M L-3 T0]
(Many a time the exponent 1 is not written explicitly, i.e, one may write [F] = [M L T-2] .)
Dimensional Analysis and its Application
Checking the Dimensional Consistency of Equations: The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. Thus, velocity cannot be added to force, or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle called the principle of homogeneity of dimensions in an equation is extremely useful in checking the correctness of an equation. If the dimensions of all the terms are not same, the equation is wrong.
Dimensions are customarily used as a preliminary test of the consistency of an equation when there is some doubt about the correctness of the equation.
Now we can test the dimensional consistency or homogeneity of the equation
for the distance x traveled by a particle or body in time t which starts from the position x0 with an initial velocity v0 at time t = 0 and has a uniform acceleration a along the direction of motion.
The dimensions of each term may be written as
[x] = [L]
[x0 ] = [L]
[v0 t] = [L T-1] [T] = [L]
[(1/2) a t2] = [L T-2] [T2]= [L]
As each term on the right-hand side of this equation has the same dimension, namely that of length, which is same as the dimension of the left-hand side of the equation, hence this equation is a dimensionally correct equation.
It may be noted that a test of consistency of dimensions tells us no more and no less than a test of consistency of units, but has the advantage that we need not commit ourselves to a particular choice of units, and we need not worry about conversions among multiples and sub-multiples of the units.
It must be remembered that if an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong.
Deducing Relation among the Physical Quantities: The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this, we should know the dependence of the physical quantity on other quantities (up to three physical quantities or linearly independent variables) and consider it as a product type of the dependency.
Dimensional analysis is very useful in deducing relations among the interdependent physical quantities.
However, dimensionless constants cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities in any equation.
It does not distinguish between the physical quantities having same dimensions.