Dimensions of physical quantities
Dimensions:
It represents the fundamental nature of a physical quantity in terms of fundamental (basic) quantities (such as mass, length, time, etc.). It helps understand and compare different physical quantities.
The dimension of a physical quantity is the powers (or exponents) to which the fundamental quantities (mass, length, time, etc.) are raised to represent that quantity.
Dimensional Formula
- Detailed expression that shows how and which of the fundamental quantities are involved in a particular physical quantity.
- Represented using square brackets with specific powers for each base quantity. \([M^a L^b T^c I^d θ^e N^f J^g]\)
Dimensional Formula
The expression of equality that equates a physical quantity with its dimensions.
| Fundamental Physical Quantity | Dimension | Dimension Formula |
|---|---|---|
| \(\textbf{[Length]}\) | \([L]\) | \([M^{0}L^{1}T^{0}] \) |
| \(\textbf{[Mass]}\) | \([M]\) | \([M^{1}L^{0}T^{0}] \) |
| \(\textbf{[Time]}\) | \([T]\) | \([M^{0}L^{0}T^{1}] \) |
| \(\textbf{[Electric Current]}\) | \([I]\) | \([M^{1}L^{0}T^{0} I^{1}] \) |
| \(\textbf{[Temperature]}\) | \([\Theta]\) | \([M^{1}L^{0}T^{0} \Theta^{1}] \) |
| \(\textbf{[Luminous Intensity]}\) | \([J]\) | \([M^{1}L^{0}T^{0} J^{1}] \) |
| \(\textbf{[Amount of Substance]}\) | \([N]\) | \([M^{1}L^{0}T^{0} N^{1}] \) |
| Supplementary Physical Quantities | Dimension Formula | |
| \(\textbf{[Plane Angle]}\) | \([M^{0}L^{0}T^{0}]\) | |
| \(\textbf{[Solid Angle]}\) | \([M^{0}L^{0}T^{0}]\) |
| Important Derived Physical Quantity | Dimension | Dimension Formula |
|---|---|---|
| \(\textbf{[Volume]}\) | \([L^{3}] \) | \([M^{0}L^{3}T^{0}] \) |
| \(\textbf{[Density]}\) | \([ML^{-3}] \) | \([M^{1}L^{-3}T^{0}] \) |
| \(\textbf{[Velocity]}\) | \([LT^{-1}] \) | \([M^{0}L^{1}T^{-1}] \) |
| \(\textbf{[Acceleration]}\) | \([L^{}T^{-2}] \) | \([M^{0}L^{1}T^{-2}] \) |
| \(\textbf{[Force]}\) | \([M^{}L^{}T^{-2}] \) | \([M^{1}L^{1}T^{-2}] \) |
| \(\textbf{[Energy]}\) | \([M^{}L^{2}T^{-2}] \) | \([M^{1}L^{2}T^{-2}] \) |
| \(\textbf{[Charge]}\) | \([T^{}I^{}] \) | \([M^{0}L^{0}T^{1}I^{1}] \) |
| \(\textbf{[Frequency]}\) | \([T^{-1}] \) | \([M^{0}L^{0}T^{-1}] \) |
Dimensional Analysis
Definition
Analysis of the relationships between different physical quantities by analyzing their dimensions in terms of fundamental physical quantities.
Applications of dimensional analysis
- Checking dimensional consistency (homogeneity) of equations
- Establishing relationship between various physical quantities (Deriving formula)
- Unit conversion
Limitations of Dimensional Analysis
- We need to know all the physical quantities that are related with a physical quantity.
- Consistency test is not perfect. A dimensionally incorrect formula is always incorrect but a dimensionally correct formula may be incorrect too.
- Cannot derive exact formulas if dimensionless constants are part of it.
- The method to get relationship between physical quantities works only if there is as many equations as unknowns.
