## Saturday, 20 July 2013

### Units and Dimensions in Physics

 This chapter deals with Units and Dimensions. If you are already aware of Fundamental, Derived units and number of dimensions of any Physical unit, you can skip this section.

The word "Unit" means one. A physical unit is required to quantitively describe a physical quantity. The measurement of a quantity is mentioned in two parts, the first part gives how many times of the standard unit and the second part gives the name of the unit.

For example - The distance of moon from the earth is 384,400 Km. Here 384,400 is the number and Km (Kilometer) is the unit.

There are three system of Units :

1. CGS : Physical quantities are expressed in Centimeters, Gram, Second.
2. FPS : Physical quantites are expressed in Foot, Pound, Second.
3. MKS : Physical quantities are expressed in Meter, Kilogram, Second. Remember this system is not the same as SI system.

Fundamental and Derived Quantites :

Those units that are fundamental by nature and cannot be broken down into more baic units are called Fundamental Units. For example Length(L), Time(T), Mass(M), Current(A), Amount of Substance(mol), Luminous Intensity(cd) and Temperature(K).

Those units which can be broken down to obtain fundamental units are called Derived Units. For example Force = Mass x Acceleration = Mass x Velocity/Time = Mass x Length / (Time x Time), i.e Force = Mass x Length / (Time x Time).

SI system of physical quantities :

A body name Conference Generale des Poind et Mesures (CGPM), also called General  Conference on Weights and Measures is responsible for maintaining International standards. In 1971 CGPM held a meeting and decided a system of units called the International System of Units. It is abbreviated as SI from the French name Le Systeme International d'Unites. The system of units is Internationally accepted.

SI Units of Fundamental Quantities

Quantity                             Name of Unit       Symbol

Length                                                Meter                  m
Mass                                                 Kilogram             Kg
Time                                                    Second               s
Current                                               Ampere              A
Amount of Substance                         Mole                  mol
Thermodynamic Temperature           Kelvin                 K
Luminous Intensity                             Candela              cd

SI Prefixes

If KG is taken as base unit, we would have to express mass of electron as 9.1 x 10-31 Kg = 9.1 / 10000000000000000000000000000000 Kg. This seems to be too complicated to deal with. So CGPM recommends using the following Prefixes before the Fundamental units

Metric prefixes in everyday use

Text Symbol Factor

tera T 1000000000000
giga G 1000000000
mega M 1000000
kilo K 1000
hecto H 100
(none) (none) 1
deci d 0.1
centi c 0.01
milli m 0.001
micro u 0.000001
nano n 0.000000001
pico p 0.000000000001

How to define these Fundamental Units

These fundamental units are defined on the basis of their ease interconvertibility to other units and easy avalability.

Meter - It is the SI unit of Length. 1m is defined as the distance traveled by the light in 1/299,792,458 second.

Kilogram - It is the SI unit of Mass. 1Kg is defined as the mass equal to the mass of cylinder made of platinum-iridium alloy kept at International Bureau of Weights.

Second - It is the SI unit of Time. Cesium 133 atom emits electromagnetic radiation of several wavelengths. A particular radiation is selected which corresponds to the transition between two hyperfine levels of ground state of Cesium 133 atom. Each radiation has a time period of repetition of certain characteristics which is exact to each other. Thus 1s is defined as the time duration in 9,192,631,770 time periods of the selected transition. Since this second is so accurate, Meter has been defined in terms of second.

Please go through the definition of second again as it would be helpful for you to understand Time Dilation. The rest of the units are for your knowledge. There would be little or no use of these units in the coming chapters.

Ampere - It is the SI unit of Current. 1A is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10-7 newton per metre of length.

Mole - It is the SI unit of Amount of Substance. 1mol is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

Kelvin - It is the unit of Thermodynamic temperature. 1K is the fraction 1 / 273.16 of the thermodynamic temperature of the triple point of water.

Candela - It is the unit of Luminous Intensity. 1cd is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Derived Units in terms of Fundamental Units.

Suppose a unit of Speed, i.e m/s. It is in terms of Meter and Second as m/s or ms-1. Now unit of work is Joules (J) = newton x meter = kilogram x meter / (second x second) x meter. So 1J=1Kg.m2/s2. As you can see that every Derived units can be converted to Fundamental units by breaking it down.

### Dimensions

Dimensions of a physical quantity is defines as the powers of the fundamental physical quantities to which the fundamental quantities are raised to get Derived quantities.

For example, Speed is in terms of meter and second, i.e Length and Time. Thus dimensional formula of Speed is [LT-1], where L stands for Length and T stands for Time and square brackets ([ ]) are used to specify Dimensional formula.

Similarly, Thermal conductivity = Work/displacement-temperature = mass x length x time-2 x length/length x temperature = [MLT-3K-1].
A complete list of dimensional formula of major physical unit is here.

No. of dimensions

You must have heard of a physical quantities being 1 dimensional, 2 dimensional, 3 dimensional etc.
No.of dimensions of a physical quantity is calculated by following method :

1. 1st express the quantitity in terms of its Dimensional formula. For example Dimenional formula of Permittivity of Space is [M-1L-3T4I2].
2. Add up the powers of the dimensions. In our example its -1-3+4+2=2.
3. Mod the addition. In our example |2|=2. Thus Perttimivity of space is a 2 dimensional physical quantity.

Similarly, Inductance=[ML2T-2I-2], 2-2+1-2=-1, |-1|=1, Thus Inductance is a one dimensional physical quantity.

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