Dimensions of Physical Quantities

Dimensions of Physical Quantities from Section 1.4 – CBSE Class 11 Physics (Units and Measurements)

📘 1.4 – Dimensions of Physical Quantities

🔍 What are Dimensions?

The dimension of a physical quantity refers to the power (or exponent) to which a base quantity must be raised to represent that physical quantity.

For any derived quantity, we express it in terms of fundamental dimensions like:

$$\text{[Length]} = [L], \quad \text{[Mass]} = [M], \quad \text{[Time]} = [T], \quad \text{[Electric Current]} = [A], \\ \text{[Temperature]} = [K], \quad \text{[Luminous Intensity]} = [cd], \quad \text{[Amount of Substance]} = [mol]$$

These seven are the fundamental or base dimensions in physics.

🔢 Representing Dimensions

The notation [Q] refers to the dimensions of a physical quantity Q.

For example:

• If $Q = \text{Force}$, then $[Q] = [M L T^{-2}]$

• If $Q = \text{Speed}$, then $[Q] = [L T^{-1}]$

📐 Examples in Mechanics

In mechanics, all quantities can be expressed using only [M], [L], [T].

📦 Example 1: Volume

Volume = Length × Breadth × Height
Each is of dimension $[L]$, so:

$$[\text{Volume}] = [L] \cdot [L] \cdot [L] = [L^3]$$

It has:

• 3 dimensions in length

• 0 in mass $[M^0]$

• 0 in time $[T^0]$

🧲 Example 2: Force

$$\text{Force} = \text{Mass} \times \text{Acceleration}$$ $$[\text{Mass}] = [M], \quad [\text{Acceleration}] = \frac{[L]}{[T^2]} = [L T^{-2}]$$ $$[\text{Force}] = [M] \cdot [L T^{-2}] = [M L T^{-2}]$$

Force has:

• 1 dimension in mass

• 1 in length

• –2 in time

🚗 Example 3: Velocity

Velocity = Displacement / Time

$$[\text{Velocity}] = \frac{[L]}{[T]} = [L T^{-1}]$$

Same applies to:

• Initial velocity

• Final velocity

• Average velocity

• Speed

Even though magnitudes vary, their dimensions remain the same.

✅ Important Characteristics of Dimensional Representation

• Only the type or nature of quantity is described.

• It does not involve magnitude or unit.

• Different quantities with same dimensional formula are called dimensionally similar.

• It helps:

  • Check dimensional consistency of equations

  • Derive relations between quantities

  • Convert units between systems

🧮 Zero Dimensions in a Base Quantity

A quantity can have zero power in a particular base quantity, meaning it doesn't depend on it.

For example:

• Volume: $[M^0 L^3 T^0]$

• Speed: $[M^0 L^1 T^{-1}]$

🚫 Not All Physical Properties Are Dimensioned

• Quantities like refractive index, strain, angle (in radians) are dimensionless.

• Still physical, but have no associated base dimension.

🧠 Summary Table

Physical Quantity	Dimensional Formula
Speed/Velocity	$[L T^{-1}]$
Acceleration	$[L T^{-2}]$
Force	$[M L T^{-2}]$
Work/Energy	$[M L^2 T^{-2}]$
Power	$[M L^2 T^{-3}]$
Pressure	$[M L^{-1} T^{-2}]$
Density	$[M L^{-3}]$
Momentum	$[M L T^{-1}]$

🧾 Conclusion

Understanding dimensions is essential for:

• Analyzing physical relationships

• Verifying equations

• Deriving new formulas

• Performing conversions across unit systems

By using dimensional analysis, we gain powerful insight into the structure of physical laws, without the need for detailed experiments at every step.