Dimensional Formulae and Dimensional Equations

Section 1.5 – Dimensional Formulae and Dimensional Equations from Class 11 Physics (Units and Measurements)

📘 1.5 – Dimensional Formulae and Dimensional Equations

🧠 What is a Dimensional Formula?

A dimensional formula is an expression that shows how and which of the base physical quantities (Mass $[M]$, Length $[L]$, Time $[T]$, etc.) are used to represent a given physical quantity.

$$\text{Dimensional Formula} = [M^a L^b T^c A^d K^e \text{mol}^f cd^g]$$

Here,

• $a, b, c, d, e, f, g$ are integers (positive, negative, or zero)

• Each represents the power of the respective base quantity

📌 Examples of Dimensional Formulae

Physical Quantity	Dimensional Formula
Volume $V$	$[M^0 L^3 T^0]$
Speed $v$	$[M^0 L T^{-1}]$
Acceleration $a$	$[M^0 L T^{-2}]$
Density $ρ$	$[M^1 L^{-3} T^0]$
Force $F$	$[M^1 L^1 T^{-2}]$
Energy $E$	$[M^1 L^2 T^{-2}]$

🧮 What is a Dimensional Equation?

A dimensional equation is formed when a physical quantity is equated to its dimensional formula.

For example:

• Volume:          $[V] = [M^0 L^3 T^0]$

• Velocity:         $[v] = [M^0 L T^{-1}]$

• Force:           $[F] = [M L T^{-2}]$

• Density:         $[ρ] = [M L^{-3} T^0]$

🧩 Deriving Dimensional Equations from Known Relations

You can derive the dimensional equation of a quantity from the physical law or formula it follows.

📌 Example 1: Force

We know:

$$F = m \cdot a$$

Where:

• Mass $m = [M]$

• Acceleration $a = [L T^{-2}]$

$$\Rightarrow [F] = [M] \cdot [L T^{-2}] = [M L T^{-2}]$$

So the dimensional equation of force is:

$$[F] = [M L T^{-2}]$$

📌 Example 2: Density

We know:

$$ρ = \frac{m}{V}$$

Where:

• Mass $m = [M]$

• Volume $V = [L^3]$

$$\Rightarrow [ρ] = \frac{[M]}{[L^3]} = [M L^{-3}]$$

So the dimensional equation of density is:

$$[ρ] = [M L^{-3} T^0]$$

✅ Why Dimensional Equations Are Useful

• 🔎 Check dimensional consistency of physical equations

• 🧩 Derive new relations between physical quantities

• 🌐 Convert units from one system to another

• 🧪 Help identify hidden physical relations in complex formulas

⚠️ Limitations

• Dimensional analysis cannot determine constants (like $\frac{1}{2}, 2\pi$, etc.)

• It cannot distinguish between scalar and vector quantities

• It fails if a physical quantity is expressed as a sum/difference of different dimensional terms

🧾 Summary Table

Quantity	Formula	Dimensional Equation
Volume $V$	$L \times B \times H$	$[V] = [M^0 L^3 T^0]$
Velocity $v$	$\frac{d}{t}$	$[v] = [M^0 L T^{-1}]$
Force $F$	$m \cdot a$	$[F] = [M L T^{-2}]$
Pressure $P$	$\frac{F}{A}$	$[P] = [M L^{-1} T^{-2}]$
Energy $E$	$F \cdot d$	$[E] = [M L^2 T^{-2}]$
Density $ρ$	$\frac{m}{V}$	$[ρ] = [M L^{-3} T^0]$

🔚 Conclusion

Dimensional formulae and dimensional equations form the backbone of unit analysis and physical reasoning in physics. They allow us to:

• Express physical quantities independent of any unit system

• Test equations for correctness

• Develop insights into physical laws

They act like a "grammar" in the language of physics, helping ensure that all expressions and equations are physically and mathematically valid.