JEE Physics Units and Measurements Notes Set 03

Physical Quantities

A physical quantity is anything that can be measured and used to state a law of physics. Quantities are of three kinds. Fundamental (base) quantities are independent of one another and all other quantities are built from them. Derived quantities are expressed in terms of the base quantities. Supplementary quantities are plane angle and solid angle.
The seven base quantities chosen internationally are length (L), mass (M), time (T), electric current (A), temperature (K), amount of substance (mol) and luminous intensity (cd). Base quantities must be independent: velocity, area or distance cannot be chosen as base quantities along with length and time, because they are formed from them.

Dimensions and Dimensional Formula

Writing a physical quantity in terms of the base quantities gives its dimensional formula, and the powers of the base quantities are its dimensions. For example, momentum is \( p=mv \), so \( [p]=[MLT^{-1}] \): one dimension in mass, one in length and minus one in time.
The dimension of a quantity does not depend on which formula is used. Numbers, angles and trigonometric, logarithmic and exponential functions are dimensionless, so the volume of a sphere \( \frac43\pi r^3 \) still has dimension \( L^3 \).
Commonly used dimensional formulae: force \( MLT^{-2} \); work, energy and torque \( ML^2T^{-2} \); power \( ML^2T^{-3} \); pressure, stress and Young's modulus \( ML^{-1}T^{-2} \); density \( ML^{-3} \); angular velocity \( T^{-1} \); angular momentum and Planck's constant \( ML^2T^{-1} \); gravitational constant \( M^{-1}L^3T^{-2} \); coefficient of viscosity \( ML^{-1}T^{-1} \); surface tension \( MT^{-2} \); specific heat \( L^2T^{-2}K^{-1} \); gas constant \( ML^2T^{-2}\text{mol}^{-1}K^{-1} \); Boltzmann constant \( ML^2T^{-2}K^{-1} \); thermal conductivity \( MLT^{-3}K^{-1} \); Stefan's constant \( MT^{-3}K^{-4} \); resistance \( ML^2T^{-3}A^{-2} \); inductance \( ML^2T^{-2}A^{-2} \); magnetic flux \( ML^2T^{-2}A^{-1} \); magnetic field \( MT^{-2}A^{-1} \); permittivity \( M^{-1}L^{-3}T^4A^2 \); permeability \( MLT^{-2}A^{-2} \).

Rules for Dimensional Homogeneity

Only quantities of the same kind can be added or subtracted, and the result is of that same kind. In \( x=a+bt \), a has the dimension of x and bt does as well.
The argument of sin, cos, log or an exponential must be dimensionless. Hence in \( F=A\sin Ct+B\cos Dx \), C has dimension \( T^{-1} \), D has \( L^{-1} \), and A and B have the dimension of force.
Both sides of a correct physical equation must have the same dimensions (principle of homogeneity).

Uses and Limits of Dimensional Analysis

Uses: (1) check whether an equation is dimensionally consistent; (2) convert a quantity from one system of units to another; (3) derive a relation between quantities when the dependence is a product of powers, by writing \( Q=k\,a^xb^yc^z \) and equating the powers of M, L and T.
Examples of derivation: the speed of a water wave depends on wavelength and g, giving \( v^2=kg\lambda \); the angular frequency of a body oscillating about a ring depends on m, r and G, giving \( \omega\propto\sqrt{Gm/r^3} \); using h, c and G as base units gives mass \( \sqrt{hc/G} \), length \( \sqrt{hG/c^3} \) and time \( \sqrt{hG/c^5} \).
Limits: it cannot give dimensionless constants such as 1/2 or 2π, cannot handle sums of terms or trigonometric and exponential relations directly, and fails when a quantity depends on more than three others in a way that cannot be separated. A dimensionally correct equation need not be physically correct.

Units and Conversion

The SI system has seven base units: metre, kilogram, second, ampere, kelvin, mole and candela. Plane angle (radian) and solid angle (steradian) are supplementary units. Derived units include newton, joule, watt, pascal, hertz, coulomb, volt, ohm, farad, henry, weber and tesla.
Common prefixes: tera 10¹², giga 10⁹, mega 10⁶, kilo 10³, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹², femto 10⁻¹⁵.
If a quantity has dimension \( M^aL^bT^c \), its numerical value in a new system is \( n_2=n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c \). Example: \( G=6.67\times10^{-11} \) N m² kg⁻² becomes \( 6.67\times10^{-8} \) in CGS units. If the units of length and time are both doubled, the numerical value of g doubles.
Some pairs share dimensions: impulse and momentum; work, energy and torque; Planck's constant and angular momentum; stress, pressure and Young's modulus; frequency, angular velocity and decay constant. Volt second is the unit of magnetic flux, and ohm times farad is a time.

Worked Examples

1. Dimensions of a in the van der Waals equation: \( \left(P+\frac{a}{V^2}\right)(V-b)=RT \) gives \( [a]=[PV^2]=ML^5T^{-2} \) and \( [b]=L^3 \), so \( [ab]=ML^8T^{-2} \).
2. Stress at the leg of a giant whose linear dimensions are 9 times larger: weight scales as \( L^3 \), area as \( L^2 \), so the stress becomes 9 times larger.
3. Fine-structure constant: \( \frac{e^2}{4\pi\varepsilon_0\hbar c} \) is dimensionless because both numerator and denominator have dimension \( ML^3T^{-2} \).