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Unit, Dimension & Error


3.1. Dimensions of a physical quantity are the powers to which the fundamental units of mass, length, time etc. must be raised in order to represent that physical quantity.
Dimensional formula = [Ma Lb Tc Qd] where a, b, c, d are the dimensions of M, L, T, Q respectively.
Some Points About Dimensions :
(a) The dimensions of a physical quantity do not depend upon system of units to represent that physical quantity.
(b) Pure numbers and pure ratio do not have any dimensions. i.e. these are dimension less, e.g. refractive index, relative density, relative permeability, cos θ, π , strain etc.
(c) Similar dimension can be added or subtracted but it does not change the dimensions.
(d) For a physical equation to be correct dimensionally the dimension of all terms on two sides of the equation must be same. This is known as the principle of homogeneity of dimensions.
(e) Logarithmic functions as log x, ex is the dimension less quantity.
(f) Powers are dimension less.
(g) If we put the value of any physical quantity in any formula it seems unbalanced but reality is that it is balanced formula. Only appearance is unbalanced as:

(h) The dimensions of two physical quantities may be same but the quantities need not be similar.
(i) Remember the following dimensional formula –
Force= [M1L1T––2]
Energy = [M1L2T–––2]
3.2.Uses of Dimension : The uses of dimension are as given below.
3.2.1 Homogeneity of dimensions in equation.
3.2.2 Conversion of units
3.2.3Deducing relation among the physical quantities.
3.2.1 Homogeneity of Dimensions in Equation:
The dimensions of all the terms in an equation must be identical. This simple principle is called the principle of homogeneity of dimensions. This is the very useful method whether an equation may be correct or not. If the dimensions of all the terms are not same the equation must be wrong. Let us check the equation.
x = ut + at2
[x] = L
[ut] = velocity × time
= × time = L
= [at2]
= acceleration × (time)2
= × (time)2
= × time2 = L
Thus the equation is correct as far as the dimensions are concerned. The equation x = ut + at2 is also dimensionally correct although this is an incorrect equation. So a dimensionally correct equation need not be physically correct but a dimensionally wrong equation must be wrong.
3.2.2 Conversion of Units :
When we choose to work with a different set of units for the base quantities, the units of all the derived quantities must be changed. Dimensions can be useful in finding the conversion factor for the unit of a derived physical quantity from one system to other.
3.2.3 Deducing Relation among the physical quantities :
Some times dimensions can be used to deduce a relation between the physical quantities. If one knows the quantities on which a particular physical quantity depends and if one is given that this dependence is of product type. Method of dimension may be helpful to derive the relation.
3.3. Limitations of the dimensional method :
(a) First of all we have to know the quantities on which a particular physical quantity depends.
(b) Method works only if the dependence is of the product type (Not applicable for x = ut + at2)
(c) Numerical constants having no dimensions can not be deduce by the method of dimensions.
(d) Method works only if there are as many equations available as there are unknowns.
3.4. Application of dimensional analysis :
(1) Write the definition or formula for the physical quantity.
(2) Replace M, L and T by the fundamental units of the required system to get the unit of physical quantity
[a] Gravitational constant G :
I. Approach
From Newton's law of gravitation we have

so its SI units is
II. Approach
From the relation between G and g we have

Substituting the dimensions of all physical quantities
so its SI units is
[b] Plank constant h:
I. Approach
According to constant h:
E = h ν

Substituting the dimensions of known physical quantities:
II. Approach
De Broglie

h = λ mv
Substituting the dimensions of known physical quantities:
[h] = [L] [M] [LT––1]
[h] = [ML2 T––1]
[h] = [ML2 T–1]
III. Approach
Bohr's II Postulate

Substituting the dimensions of known physical quantities: [h] = [mvr] as 2π and n are dimensionless.
⇒ So SI unit of plank's constant is kg m2/s. Which can also be written as (kg m2/s2) × s. But as kg m2/s2 is Joule so unit of h is Joule × sec. i.e. J-s
[c] Coefficient of Viscosity η
I. Approach
According to Newton's law

Substituting the dimensional formulae of all other known physical quantities.

II. Approach
Stoke's law F = 6 π η rv

Substituting the dimensional formulae of all other known physical quantities.

III. Approach
Poiseuille's formula

Substituting the dimensional formulae of all other known physical quantities.

So SI or MKS unit of coefficient of viscosity is kg/m-s or (g/cm-s called poise in C.G.S. system)
3.4.2 IN HEAT :
[a] Temperature : In heat it is assumed to be a fundamental quantity with dimensions [θ] and unit Kelvin [K]. [How ever, ΔK = ΔC°].
[b] Heat : It is energy so it dimensions are [ML2T––2] and SI Units Joule (J). Practical unit of heat is calorie (cal.) and 1 calorie = 4.18 joule
[c] Coefficient of Linear Expansion a :
It is defined as

i.e.[α] = [θ –1] so its unit is (C°)––1 or K–1
[d] Specific Heat C

[C] = [L2 T–2 θ ––1]
So its SI unit be J/kg-K while practical unit cal/gm-C°
[e] Latent Heat L
By definition
Q = ML
i.e. [L] = [ML2 T––2] / [M]
⇒ [L] = [L2 T––2]
So its SI unit be J/kg while practical unit cal/gm
[f] Coefficient of Thermal Conductivity K
According to law of thermal conductivity

⇒[K] = [MLT––3 θ––1]
Its SI unit is W/mK while practical unit is cal/s-cm-C°.
[g] Mechanical Equivalent of Heat J :
According to I law of thermodynamics
W = JH

i.e.[J] = [M0 L0 T0]
i.e. J has no dimensions. Its practical unit is J/cal and has value 4.18J/cal.
[h] Boltzmann constant K: According to kinetic theory of gases, energy of a gas molecule is given by

i.e. [K] = [ ML2T––2 θ––1]
So its SI unit is J/K and value 1.38 × 10––23 J/K.
[i] Gas constant R:
According to gas equation for perfect gas
PV = μRT

i.e. [R] = [ML2 T––2 θ––1 μ––1]
So its SI unit is J/mol-K. While practical unit is cal/mol-K. It is a universal constant with value 8.31 J/mol-K or 2 cal/mol-K.
[j] Vander Waal’s constants a and b :
Vander Waal's equation

Vander Waal'’s equation for μ mol is –

compare equation (1) and (2)
μ2 a' = a
and μ b' = b

⇒ [a'] = [mL5 T––2 μ–2]
and [b'] = [L3 μ–1]
Unit of a' and b' are respectively.
(a)Current I : While dealing electricity we assume current to be a fundamental quantity and represent it by [A] with unit ampere (A)
(b) Charge Q
AsI =
So [Q] = [I] [t]
⇒ [Q] = [At]
The SI unit of charge is A × s and is called coulomb (C).
Note :
(i) In MKSQ system charge is assumed to be fundamental quantity with dimension [Q] and unit coulomb. So in this system current will be derived with dimension [QT––1] and units coulomb / s which is ampere.
(ii)In CGS system there are two units of charge namely esu of charge frankline (Fr) and emu of charge. It is found that
1 coulomb = 3 × 109 esu of charge
= emu of charge.
(c) Electric potential V :
It is defined as

i.e. [V] = [ML2 T–3 A––1]
So SI unit of potential is J/C and is called volt (V)

(d) Electric intensity E :
It is defined as

(e) Capacitance C
It is defined as
q = CV

and its unit coulomb / volt is called farad.
Note : as W = qV joule / coulomb is volt → V, q = CV coulomb/volt is farad → F
(f) Permittivity of free space ε0 :
According to coulomb's law

(g) Dielectric constant K or Relative permittivity εr As K → εr = (ε/ε0) so it has no units and dimensions.
(h) Resistance R:
According to ohm's law V = IR

and its unit volt / ampere is called ohm (Ω).
(i) Resistivity or specific Resistance ρ :

and its unit is ohm-m or ohm-cm.
(j) Coefficient of self induction L :
According to definition of

(k) Magnetic flux φ :
According to faraday's law of electro-magnetic induction

⇒ [φ] = [ML2 T–2/AT] [T]
= [ML2 T––2 A–1]
and its unit will be volt × s known as Weber (Wb.)
(l) Magnetic Induction B :
As force on a current element in a magnetic field is given by F = Biλ sinθ

(m) Magnetic Intensity H :
For Biot-savart law :

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