# Units and Dimensions

The fundamental quantities characterize different classes or groups of physical quantities, irrespective of the units system applied for measurement and their quantitative values. Put differently, a fundamental quantity refers to a class of physical quantities of its own type where each one of them has no dependence on any other fundamental quantities. The attribute which is common to a class of physical quantities is known as their dimensionality.

Many physical quantities are expressed in the form of combinations of five dimensions:

• Length (L)
• Mass (M)
• Time (T)
• Current (I)
• Temperature (oC or oF)

These five dimensions are known to be the basic since they are easy to measure. Dimensions are not similar to units. For instance, the physical quantity like speed can be measured in units of kilometers per hour, but speed is a length which is divided by time, hence the dimensions of speed are length divided by time.

The dimension of a physical quantity is the power to which the fundamental units are raised to express the quantity. Dimension deals with qualitative part of a measurement.

## Dimensional Formula and Dimensional Equation

A dimensional formula of a physical quantity is the formula that shows which are the fundamental units applied for the measurement of that quantity.

The equation shown below written in the following manner is known as a dimensional equation.

Area = [M0L2T0]

### How to Write Dimensions of Physical Quantities?

Dimension is determined in the following manner:

1. Write the formula with the quantity towards the left-hand side of the equation.
2. Convert all the quantities into fundamental quantities of length, mass and time.
3. Substitute L, M, and T for length, mass and time respectively.

## Base quantities and their Units:

 Base quantity Unit Symbol Length Meter M Mass Kilogram Kg Time Second Sec Electric current Ampere A Temperature Kelvin K Luminous intensity Candela Cd Amount of substance Mole Mole

## Dimensions:

The unit of derived quantities is based on one or more fundamental units, expressed by the help of dimensions of that derived unit. Put differently, the dimensions of a physical quantity describe how its unit is relevant to the fundamental units.

Every fundamental unit is expressed by a capital letter to express dimensions. Hence L denoted the unit of length, M denotes a unit of mass, T denotes a unit of time and I denotes a unit of current.

## Derived SI Units and  Symbols:

 Quantity Unit Symbol Express in base units Force newton N Kg-m/sec2 Work joules J Kg-m2/sec2 Power watt W Kg-m2/sec3 Pressure Pascal Pa Kg m-1/S2

## Characteristics of Dimensions:

Dimensions of any physical quantity are independent of unit systems.

Quantities that have identical dimensions can be added or subtracted from each other.

Two different physical quantities can have similar dimensions.

Units of a physical quantity can be obtained from its dimensions and vice-versa.

Division or multiplication of two dimensions results in the production of dimensions of a third quantity.

#### Practise This Question

A current I is passing through a wire having two sections P and Q of uniform diameters d and d2 respectively.  If the mean drift velocity of electrons in sections P and Q is denoted by vpand vQrespectively, then