Units and Dimensions in Physics Continued
The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity show its until its related to the fundamental units.
To express dimensions, each fundamental unit is represented by a capital letter. Thus, the unit of length is denoted by L, unit of mass by M, unit of time by T, unit of electric current by I, unit of temperature by K and unit of luminous intensity by C. Generally dimensions are expressed in terms of charge Q, instead of current I. The quantities in mechanics depend only upon the units of length, mass and time.
Dimension of a physical quantity can be defined as algebraic expression giving explicit relation of that physical quantity to the fundamental quantities. The dimensions are normally written in a square bracket.
Determination of dimensions
For determining the dimensions of a physical quantity, we should know its definition or how this physical quantity is related to other fundamental quantities. The quantity is expressed in terms of it immediately related quantities and now we try to express each of these quantities in its still simpler forms. This process is continued until we get all the quantities in terms of M, L, T, Q and K. Now, the powers of all like quantities are determined and the dimensions of the physical quantity is written down.
Generally the dimensions of mechanical quantities are easier to determine since the dimensions of most of them constitute quantities readily known and consist only of M, L and T. As illustrating example let us determine the dimensions of the gravitational potential and electric capacitance.
Gravitational potential: Gravitational potential at a point in the gravitational field is the potential energy per unit mass at that point.
Vg = Gravitational Potential
= Gravitational potential energy/ Mass
= (Force) (Displacement)/Mass
= (Mass) (Acceleration) (Displacement)/Mass
= (L1 T-2) (L1) = [L2 T-2]
Therefore, Dimension of Vg = [L2T-2]
Electric Capacitance: The electric capacitance of a conductor is the ratio of the electric charge on it to the electric potential of it due to the charge
= Electric charge/ Electric potential
Now, electric potential = Work/Charge
= (Force) (displacement)/Charge
= (M1L1T-2) (L1)/Q1
Therefore C = [Q]/[M1L2T-2Q-1]
= [M-1 L-2 T2 Q2]
Uses of Dimensions
a. To verify the correctness of a physical equation: According to the principle of homogeneity of dimensions, a physical equation is correct, only if the dimensions of all the terms on both sides are the same. This is because addition and subtraction of physical quantities of same nature only are permitted and only similar quantities can be compared with each other.
b. To derive the relation between physical quantities: We can find out the form of the equation, if we know the various factors on which it depends. Such physical factors are known as ‘perimeters’.
c. To convert the unit of a physical quantity from one system to another system: example of this can be,
Force = [M1L1T-2],
Newton = (kg)1 (m)1 (S)-2
Newton/Dyne = (kg/g)1 (m/cm)1 (s/s)-2
= (10)3 (10)2 = 105
1 Newton = 105 dyne
Limitations of Dimensional Analysis
Although dimensional analysis is very useful it cannot lead us too far as,
If dimensions are given, physical quantity may not be unique as many physical quantities have some dimensions. For example if the dimensional formula of a physical quantity is [ML2T-2] it may be work or energy or torque.
Numerical constant having no dimensions (K) such as (1/2), 1 or 2π etc. cannot be deducted by the methods of dimensions.
The method of dimensions cannot be used to derive relations other than product or power functions.
For example, s = ut + (1/2) at2 or y = a sin ωt cannot be derived by using this theory. However, the dimensional correctness of these can be checked.
The method of dimensions cannot be applied to derive formula if in mechanics a physical quantity depends on more than 3 physical quantities as then there will be less number (=3) of equations than the unknowns (>3). However, still we can check correctness of the given equation dimensionally. For example, T = 2 π√1/mgl cannot be derived by theory of dimensions but its dimensional correctness can be checked.
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