Chapter 1 – Units and Dimensions (JEE Physics)
1. Physical Quantity
A physical quantity is any quantity that can be measured and expressed
by a numerical value and a unit.
Examples:
- Length
- Mass
- Time
- Velocity
2. Fundamental and Derived Quantities
Fundamental Quantities
These are independent physical quantities which do not depend on others.
| Quantity | Symbol |
|---|---|
| Length | L |
| Mass | M |
| Time | T |
| Electric Current | I |
| Temperature | K |
| Luminous Intensity | cd |
| Amount of Substance | mol |
Derived Quantities
Derived quantities are obtained by combining fundamental quantities.
Examples:
- Velocity
- Force
- Energy
- Pressure
3. System of Units
A system of units is a complete set of units used for measurement.
SI System (International System)
| Quantity | SI Unit |
|---|---|
| Length | metre (m) |
| Mass | kilogram (kg) |
| Time | second (s) |
| Current | ampere (A) |
| Temperature | kelvin (K) |
JEE questions strictly follow SI units unless stated otherwise.
4. Supplementary Units (Historical)
Earlier, plane angle and solid angle were supplementary units.
- Plane angle → radian
- Solid angle → steradian
5. Dimensions
Dimensions represent the physical nature of a quantity in terms of
fundamental quantities.
Dimensions of length = $[L]$
Dimensions of mass = $[M]$
Dimensions of time = $[T]$
Dimensions of mass = $[M]$
Dimensions of time = $[T]$
6. Dimensional Formula
Dimensional formula shows how a physical quantity depends on
fundamental quantities.
Velocity = $[LT^{-1}]$
Force = $[MLT^{-2}]$
Energy = $[ML^2T^{-2}]$
Force = $[MLT^{-2}]$
Energy = $[ML^2T^{-2}]$
7. Dimensional Constants
Quantities having dimensions are called dimensional constants.
Examples:
- Gravitational constant
- Planck’s constant
8. Dimensionless Quantities
Quantities having no dimensions.
Examples:
- Angle
- Strain
- Refractive index
- Coefficient of friction
9. Principle of Dimensional Homogeneity
In any physical equation, dimensions on both sides must be equal.
This principle is used to:
- Check correctness of equations
- Derive relations
- Convert units
10. Checking Dimensional Correctness
An equation is dimensionally correct if LHS and RHS have same dimensions.
Example:
$s = ut + \frac12 at^2$
All terms have dimensions of length.
11. Derivation of Formula Using Dimensions
Suppose $y$ depends on $a, b, c$:
$y = k a^x b^y c^z$
Dimensional analysis is used to find powers $x, y, z$.
12. Limitations of Dimensional Analysis
- Cannot determine numerical constants
- Cannot derive equations with trigonometric functions
- Cannot distinguish between scalar and vector quantities
13. Conversion of Units
If a quantity has dimensional formula $[M^aL^bT^c]$:
$n_1 u_1 = n_2 u_2$
Conversion depends on dimensions.
14. Significant Figures
Significant figures indicate accuracy of measurement.
Rules:
- All non-zero digits are significant
- Zeros between digits are significant
- Trailing zeros without decimal are not significant
15. Rounding Off
Rounding is done based on the digit following the last significant digit.
16. Errors in Measurement
Error = Measured value − True value
Types:
- Absolute error
- Relative error
- Percentage error
17. Propagation of Errors
If $Z = A^m B^n$,
$$\frac{\Delta Z}{Z} = m\frac{\Delta A}{A} + n\frac{\Delta B}{B}$$
18. Dimensional Analysis in JEE
Very frequently asked:
- Check equation correctness
- Find missing power
- Convert units
19. Common JEE Mistakes
- Forgetting SI units
- Wrong dimensional formula
- Ignoring dimensionless quantities
20. Final Revision Checklist
You have mastered this chapter if you can:
- Write dimensional formula of any quantity
- Check equations dimensionally
- Convert units correctly
- Handle errors and significant figures