Dimensions of Physical Quantities
In physics, every physical quantity can be represented by its dimensional formula. This dimensional formula expresses the quantity in terms of the basic quantities (like mass, length, time) and their powers. It helps us understand the relationship between different physical quantities and serves as a powerful tool in solving various problems in physics.
Definition
The dimension of a physical quantity refers to its expression in terms of the fundamental physical quantities: Mass (M), Length (L), and Time (T). These fundamental quantities form the base units of measurement. The dimensional formula of a physical quantity represents the powers to which these base units are raised to express the given quantity.
Dimension formula
The dimensional formula of any physical quantity is expressed as the product of the powers of the fundamental units:
Where:
- \( M \) = Mass
- \( L \) = Length
- \( T \) = Time
- \( a, b, c \) are the powers to which the respective base quantities are raised to represent the physical quantity.
Examples of Dimensional Formula
- Dimension of Force: \( [F] = [M L T^{-2}] \)
- Dimension of Energy: \( [E] = [M L^2 T^{-2}] \)
- Dimension of Power: \( [P] = [M L^2 T^{-3}] \)
Uses of Dimensions
Dimensional analysis is a fundamental tool in physics. Below are some of the primary uses of dimensional formulas:
1. To Check the Correctness of a Given Physical Equation
By ensuring that both sides of an equation have the same dimensional formula, we can confirm whether the equation is dimensionally correct or not. If the dimensions on both sides do not match, the equation is not physically valid.
2. Derivation of Formula
Dimensional analysis helps in deriving equations for physical quantities without requiring a full understanding of the underlying physical theory. For example, the dimensional formula for gravitational acceleration \( g \) can be derived using dimensional analysis.
3. Conversion Between Different Units
Dimensional analysis is extremely useful when converting a physical quantity from one system of units to another. Since the product of the numerical value and the unit of any physical quantity remains constant, dimensional analysis helps in maintaining consistency across different measurement systems.
Where:
- \( Q_1 \) and \( Q_2 \) are the physical quantities in two different systems.
- \( n_1 \) and \( n_2 \) are the numerical values of \( Q_1 \) and \( Q_2 \) respectively.
- \( [M_1 L_1 T_1] \) and \( [M_2 L_2 T_2] \) are the dimensional formulas in the respective systems.
- \( a, b, c \) are the exponents for mass, length, and time in the system.
Dimensional Analysis for Conversion
For example, if a physical quantity has the dimensional formula \( [M^a L^b T^c] \) in one system, the value of that quantity in another system with different base units (like SI, CGS, etc.) can be derived using dimensional analysis.
Dimensions of Important Physical quantities
| Physical Quantity | SI Unit | Dimensional Formula |
|---|---|---|
| Power | Watt (W) | [ML2T-3] |
| Pressure, Stress, Coefficient of Elasticity (ρ, σ, η) | Pascal (Pa) or Nm-2 | [ML-1T-2] |
| Frequency, Angular Frequency | Hz or s-1 | [T-1] |
| Angular Momentum | kg m2 s-1 | [ML2T-1] |
| Torque | Nm | [ML2T-2] |
| Gravitational Constant (G) | N m2 kg-2 | [M-1L3T-2] |
| Moment of Inertia | kg m2 | [ML2] |
| Acceleration, Acceleration due to gravity | m s-2 | [LT-2] |
| Force, Thrust, Tension, Weight | Newton (N) | [MLT-2] |
| Linear Momentum, Impulse | kg m s-1 or Ns | [MLT-1] |
| Work, Energy, KE, PE, Thermal Energy, Internal Energy | Joule (J) | [ML2T-2] |
| Surface Area, Area of Cross-Section | m2 | [L2] |
| Electric Conductivity | S m-1 | [M-1L-3T3A2] |
| Young’s Modulus, Bulk Modulus | Pa | [ML-1T-2] |
| Compressibility | m2 N-1 | [M-1L1T2] |
| Magnetic Flux | Weber (Wb) | [ML2T-2A-1] |
| Magnetic Flux Density (σ) | Wb / m2 | [MT-2A-1] |
| Intensity of a Wave | W m-2 | [MT-3] |
| Photon Flux Density | m-2 s-1 | [L-2T-1] |
| Luminous Energy | Lm s | [ML2T-2] |
| Luminance | Lux | [MT-3] |