Formula of Dimensions

Dimension is a mathematical term representing the length, breadth, or height and similar aspects of a physical quantity in a particular direction. Mathematically, it is a line segment with a direction. Multiple units are raised to powers in order to obtain dimensions for a physical quantity.

The dimensional formula is the expression in which the powers to which the basic units of a quantity are raised are represented. The units can be anything like the Mass, velocity, or the length and breadth of a quantity.

The dimensional formula is used to derive the dimensional equations.

Units of Dimensions

Each physical quantity is attached to a unit which helps add a distinct value to the number. For example, if the height of the wall is 15 and the distance between the two cities is 15, are these two quantities the same? To avoid search ambiguity and confusion, units are designed.

We can say that the height of the wall is 15 meters and the distance between the two cities is 15 Kilometers. Hence, the ambiguity is resolved when we get a proper idea of the measure of the physical quantities. Further, the dimensional formula is used to assign powers to these units.

There are multiple systems of units.

The two most popular systems of units are given below:

• The S.I. system of units

• The C.G.S. system of units

The S.I. system is more popular and used widely around the world.

Let’s take a look at a few units on which we will be performing the dimensional analysis and applying the dimensional formula. These units belong to the S.I. system of units.

• Kilogram

Kilograms are used for the measurement of the weights of items. If the quantity is too small, then we measure it in grams, and if it is too large, then we measure it as tons. They are both dependent on kilograms.

• Meter

Meter is used for the measurement of height or length and breadth of entities. One meter is the value of 1650763.73 times the wavelength of light. The light emitted is by an isotope of Krypton in a vacuum.

• Second

Second is a unit used for the measurement of time. We usually measure time in terms of hours and minutes. One minute is equal to 60 seconds and 60 minutes is equal to one hour.

• Mole

Mole is an important unit in chemistry. It is used to measure how many atoms of a particular element are equal to 12 × 10-3 kg of the element carbon.

The Dimensional Formula

The dimensional formula is a general equation used for determining to what power a specific unit of measure if physical quantity is to be raised. It can be applied with basic physical units as well as the derived units, which contain multiple fundamental units.

Suppose Q is any physical quantity made up of multiple fundamental quantities, then the dimensional formula for Q is given below:

Q = MaLbTc

Here the M represents the mass of the physical quantity, the length is represented by L, and T represents the time unit.

a, b, and c are the powers to which these units are raised.

This is because mass, time, and length are the most basic physical quantities from which the other physical quantities are derived.

Sometimes, we can also find I in the dimensional formula equation, which represents the electrical current.

Let’s apply the dimensional formula on the fee of the units of physical quantities that we discussed above:

• Length

The length is usually measured in meters. And since it has only the length aspect, and the mass and time aspects are absent, the distance formula for the length is L1M0T0 which equates to L1, which is just L.

• Mass

The general unit which is used for the measurement of mass is the kilogram. Here too, only the mass aspect exists, and the time and length aspects are absent. Hence, the dimensional formula for Mass is L0M1T0 which equates to M1, which is just M.

• Time

The S.I. unit for time is seconds. Time does not have a mass or length dimension. Hence, the dimensional formula for time will have coefficients of length, and the mass dimension is 0. So, for a time it becomes L0M0T1  which equates to T1, which is just T.

These were the basic and fundamental units with only one dimension and other dimensions with zero coefficients. Let’s take an example of a derived unit of a physical quantity.

• Density

The unit of measurement for density is mass multiple by volume. Hence, the coefficient of mass will be one, and the length will be -3.

Hence the dimensional formula for density will be

M1L-3T(0) which further becomes M1L-3.

Similarly, you can form dimensional equations for other physical quantities with the dimensional formula.

Conclusion

A unit is attached to any number in order to represent the quantity of some entity. Units can be fundamental as well as derived. The derived units are made up of fundamental units raised to different powers. These powers are identified and calculated with the help of the dimensional formula.

The dimensional formula for any physical quantity Q is given as Q = MaLbTc. Here the M, T, and L represent Mass, Length, and Time respectively, while a, b and c are the powers to which they are raised.

The dimensional formula can also be used for the conversion of a system of units.