Recapitulation of Dimensions and Dimensional Formula of Physical Quantities

The number of substances is measured with the help of physical quantities. For example, speed is measured in metres per second, that is, m/s, which is the S.I. unit. It helps us convey the amount of distance travelled in a single second. The unit of luminous intensity is candela. Other physical quantities derived from physical quantities can be defined with the help of the base quantities. The units of physical quantities have dimensions. The dimension of any physical quantity is the power to which the base quantities or the fundamental quantities are raised. They help perform dimensional analysis.

Dimensions and dimensional formula

A dimension is a mathematical phrase that describes the length, width, height, luminous intensity or the amount of substance of a physical quantity in a particular direction. It is a line segment with a direction in mathematics. Multiple units of base quantities are raised to powers to acquire dimensions for a physical amount.

The dimensional formula is an equation that represents the powers to which a quantity’s basic units are elevated. The units could be anything from velocity to mass to the length and width of a quantity.

The dimension equations are derived using the dimensional formula. The dimensional formulas for any physical quantity aid in calculating its dimensional equations. The dimensional formula can be presented as:

If Q represents a physical quantity, the dimensional formula for the unit will be:

Q = MaLbTc

This is the generic dimensional equation or formula.

Here M represents the mass of the physical quantity, T represents the time aspect of the physical quantity, L is the length of the physical quantity, and M, L and T are base quantities.

And a, b and c are the powers to which M, L and T are raised.

Any physical quantity’s dimensional equation will include the dimensional formula for the same physical quantity. The right-hand dimensional formula is equated to the left-hand dimensional formula to obtain the dimensional equation of the physical quantity.

Dimensional equations from dimensional formula

Let’s use the dimensional formula to calculate the dimensional equations for some of the physical quantities like length, luminous intensity, etc.

• Length

Length is commonly expressed in metres. The dimensional formula for length is L1M0T0, which is just L because it only has the length component and no mass or time aspects.

• Luminous Intensity

The S.I. unit of the luminous intensity is candela. This is a case where a physical quantity cannot be expressed in base quantities like the length, breadth and mass since luminous intensity does not have any length, breadth or mass aspect. Its dimensional equation is M0L0T0.

• Angular displacement

Angular displacement is measured in radians. It is impossible to express it in fundamental units such as time, length or mass like the luminous intensity. As a result, the dimensional formula will be equal to 0 in terms of time, length and mass. Hence, the angular displacement’s dimensional equation becomes M0L0T0.

• Density:

The formula for calculating density is mass multiplied by volume. Therefore, the mass’s coefficient will be one; the length will be -3. As a result, the dimensional density formula is M1L-3T0, which becomes M1L-3.

The dimensional formula and the dimensional equations help perform the dimensional analysis of physical quantities. Let’s take a look at what dimensional analysis is.

Dimensional analysis

In mathematical terms, the form of dimensional analysis analyses what relationships exist between physical quantities. 

Units such as length, electric current, time and mass exist in the dimensional equation of the physical quantity. These base quantities are raised to some power.

Let’s take the concept of acceleration as an example. For a moving object, M0L1T-2 will be the dimensional equation for acceleration. Meanwhile, M0L1T-1 will be the dimensional formula for velocity. By comparing the two equations, we can see that the power of mass is the same in both, as is the power of length.

Conclusion

Units can be both fundamental and derived. The derived units are made up of fundamental units. With the dimensional formula, these powers are recognised and calculated. Any physical quantity Q has a dimensional formula Q = MaLbTc. Letters M, T and L stand for Mass, Length and Time, respectively, and the letters a, b and c stand for the powers to which they are elevated. Dimensional formulas can convert a system of units.