Unit and Dimensions
In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.
Dimensions:
Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.
Fundamental and Derived Quantities
- The quantities that are independent of other quantities are called fundamental quantities. The units that are used to measure these fundamental quantities are called fundamental units. There are four systems of units namely C.G.S, M.K.S, F.P.S, and SI.
- The quantities that are derived using the fundamental quantities are called derived quantities. The units that are used to measure these derived quantities are called derived units.
Fundamental and supplementary physical quantities in SI system:
| Fundamental Quantity | System of units | ||
| C.G.S. | M.K.S. | F.P.S. | |
| Length | centimeter | Meter | foot |
| Mass | gram | Kilogram | pound |
| Time | second | Second | second |
| Physical quantity | Unit | Symbol |
| Length | Meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Intensity of light | candela | cd |
| Quantity of substance | mole | mol |
Supplementary Quantities:
| Plane angle | radian | rad |
| Solid angle | steradian | sr |
- Meter: A meter is equal to 1650763.73 times the wavelength of the light emitted in vacuum due to electronic transition from 2p10 state to 5d5 state in Krypton-86. But in 1983, 17th General Assembly of weights and measures adopted a new definition for the meter in terms of velocity of light. According to this definition, a meter is defined as the distance traveled by light in vacuum during a time interval of 1/299, 792, 458 of a second.
- Kilogram: The mass of a cylinder of platinum-iridium alloy kept in the International Bureau of weights and measures preserved at Serves near Paris is called one kilogram.
- Second: The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atoms is called one second.
- Ampere: The current which when flowing in each of two parallel conductors of infinite length and negligible cross-section and placed one meter apart in vacuum, causes each conductor to experience a force of 2 × 10-7 newtons per meter of length is known as one ampere.
- Kelvin: The fraction of 1/273.16 of the thermodynamic temperature of the triple point of water is called Kelvin.
- Candela: The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m2 at the temperature of solidifying platinum under a pressure of
101325 Nm-2 is known as one candela. - Mole: The amount of a substance of a system which contains as many elementary entities as there are atoms in 12 × 10-3 kg of carbon-12 is known as one mole.
- Radian: The angle made by an arc of the circle equivalent to its radius at the center is known as radian. 1 radian = 57o17l45ll.
- Steradian: The angle subtended at the center by one square meter area of the surface of a sphere of radius one meter is known as steradian.
- System of Units:-
(a) C.G.S (Centimeter-Grand-Second) system.
(b) F.P.S. (Foot-Pound-Second) system.
(c) M.K.S. (Meter-Kilogram--Second) system.
(d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit.
- Dimensional Formula:-
Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity.
- How to write dimensions of physical quantities:-
(a) Write the formula for that quantity, with the quantity on L.H.S. of the equation.
(b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time.
(c) Substitute M, L and T for mass, length and time respectively.
(d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively.
- Characteristics of Dimensions:-
(a) Dimensions of a physical quantity are independent of the system of units.
(b) Quantities having similar dimensions can be added to or subtracted from each other.
(c) Dimensions of a physical quantity can be obtained from its units and vice-versa.
(d) Two different physical quantities may have same dimensions.
(e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity.
Principle of homogeneity:-
It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.”
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