Unit and Measurement - Dimensions and Dimensional Analysis (Applied Physics-1)

Fundamental Concepts: Dimensions and Dimensional Formulation

Dimension: The dimensions of a physical quantity are the powers to which the fundamental quantities are raised to represent that quantity. They are denoted by square brackets around the quantity.
Dimensional formula: The expression that shows how and which base quantities represent the dimensions of a physical quantity. For a quantity A that depends on mass M, length L, time T with powers a, b, c, the dimensional formula is [A]=[M^{a}L^{b}T^{c}]
Dimensional equation: An equation obtained by equating a physical quantity with its dimensional formula, e.g., if A has dimensions described by the formula, then the dimensional equation is [A]=[M^{a}L^{b}T^{c}]
Dimensional analysis helps verify consistency and deduce relationships between physical quantities, but it does not provide actual numerical constants unless the problem yields a dimensionless constant.

Seven base quantities and their standard symbols/dimensions:
- Length:
- Symbol: [L]
- Mass:
- Symbol: [M]
- Time:
- Symbol: [T]
- Electric current:
- Symbol: [A]
- Thermodynamic temperature:
- Symbol: [K]
- Luminous intensity:
- Symbol: [cd]
- Amount of substance:
- Symbol: [mol]
If a quantity A depends on mass, length, and time with powers a, b, c, then the dimensional formula is:
- [A] = [M^{a}L^{b}T^{c}]
Example dimensional equations:
- Volume: [V] = [L^{3}]
- Speed/velocity: [v] = [L T^{-1}]
- Force: [F] = [M L T^{-2}]
- Mass density: [\rho] = [M L^{-3}]
Common derived quantities:
- Acceleration: [a_c] = [L T^{-2}]
- Impulse: [I] = [M L T^{-1}]
- Work/Energy: [W] = [M L^{2} T^{-2}]

Dimensionless quantities/arguments:
- The arguments of trigonometric, logarithmic, and exponential functions are dimensionless.
- A pure number (unitless quantity) is dimensionless.
- Ratios of similar quantities are dimensionless, e.g., angle (radians) = length/length, refractive index = \frac{c}{v}, etc.
- Plane angle and solid angle are dimensionless.
Dimensionless constants: constants with no dimensions (e.g., refractive index, relative density, velocity of light in vacuum).
Dimensional constants: constants that carry dimensions (e.g., universal gravitational constant G, electric permittivity \varepsilon_0, coefficient of elasticity). Dimensions must be carried through when used in equations.

Principle of homogeneity of dimensions: A valid physical equation must have the same dimensions on both sides.
Checking dimensional consistency:
- Only quantities with the same dimensions can be added or subtracted.
- If dimensions on both sides match, the equation is dimensionally consistent (though not necessarily correct physically).
Example (kinetic energy form):
- Expression: \tfrac{1}{2}mv^{2}
- Dimensions: Left-hand side (LHS) [E] = [M L^{2} T^{-2}]; Right-hand side (RHS) has the same dimensions, confirming dimensional consistency for the kinetic energy term.

Example 1: Dimensional check for the classic energy relation E = m c^{2}.
- Dimensions: [E] = [M L^{2} T^{-2}], [m c^{2}] = [M] [L^{2} T^{-2}] = [M L^{2} T^{-2}]
- Conclusion: Consistent.
Example 2: Wave relation v = f λ.
- Dimensions: [v] = [L T^{-1}], [f][\lambda] = [T^{-1}][L] = [L T^{-1}]
- Conclusion: Consistent.
Example 3: Power as rate of doing work: P = F v.
- Dimensions: [F][v] = [M L T^{-2}] [L T^{-1}] = [M L^{2} T^{-3}] = [P]
- Conclusion: Consistent.
Example 4: Pendulum period (dimensional deduction):
- Suppose T depends on mass m, length l, gravity g: T ∝ m^{α} l^{β} g^{γ}.
- Dimensions: [T] = [M^{0} L^{0} T^{1}] = [M^{α}][L^{β}][(L T^{-2})^{γ}] = [M^{α} L^{β+γ} T^{-2γ}].
- Equating exponents: α = 0, β + γ = 0, -2γ = 1 ⇒ γ = -1/2, β = 1/2.
- Therefore: T ∝ l^{1/2} g^{-1/2} = \sqrt{\frac{l}{g}} with a dimensionless constant K: T = K \sqrt{\frac{l}{g}}.

How to deduce a relation using dimensions:
- Identify all factors on which the quantity depends.
- Assign unknown exponents to these factors, set up the dimensional equation, and solve for the exponents by matching powers of M, L, T.
- The resulting relation is determined up to a dimensionless constant.
Example: Centripetal force F for a particle of mass m moving with velocity v in a circle of radius r.
- Suppose F ∝ m^{a} v^{b} r^{c}.
- Dimensions: F ∼ M L T^{-2}; m^{a} ∼ M^{a}; v^{b} ∼ (L T^{-1})^{b}; r^{c} ∼ L^{c}.
- Equate: M: a = 1; L: b + c = 1; T: -b = -2 ⇒ b = 2 ⇒ c = -1.
- Result: F ∝ m v^{2} / r, i.e., F = K (m v^{2} / r) with a dimensionless constant K.
Simple pendulum (Time period): Derived above: T = K \sqrt{\frac{l}{g}}.
Example 3 (escape velocity dimension): If v depends on g and R (planet parameters), set v = K g^{α} R^{β} and solve for α, β to obtain v = K \sqrt{g R}.

Fundamental SI units: mass = kg, length = m, time = s.
Corresponding CGS units: mass = g, length = cm, time = s.
Key conversion formulas:
- Length: 1 m = 100 cm
- Mass: 1 kg = 1000 g
- Time: 1 s = 1 s (unchanged)
- Force and energy conversions:
- 1 N = 10^{5} dyne
- 1 J = 10^{7} erg
Practical conversions to SG units on a problem base (example conventions):
- The dimensions of force are [F]=[M L T^{-2}]; convert between SI and CGS using base unit changes.
- Example standard result: 1 N = 10^{5} dyne, 1 J = 10^{7} erg, 1 W = 10^{7} erg s^{-1}.
Example: Gravitational constant G in CGS vs SI:
- In SI: G = 6.67\times 10^{-11} \ \mathrm{N\,m^{2}\,kg^{-2}}.
- In CGS: G = 6.67\times 10^{-8} \ \mathrm{dyne\,cm^{2}\,g^{-2}}.

Convert units between SI and CGS systems (e.g., dyne to newton, erg to joule) using dimensional analysis and base-unit conversions.
Check dimensionally the validity of proposed formulas for kinetic energy and other expressions.
Derive expressions for physical quantities using the method of dimensional analysis (e.g., energy in SHM, natural frequency relations, etc.).
Be aware of when dimensional analysis cannot be used:
- It cannot determine dimensionless constants.
- It cannot handle quantities dependent on trigonometric or exponential functions in a way that fixes constants.
- It becomes unreliable if the quantity depends on more than three independent dimensional variables or on sums/differences of terms with different dimensions.

Dimensional analysis does not provide information about dimensionless constants.
It cannot be applied to quantities that depend on trigonometric or exponential functions in a way that requires specific constants.
It may be difficult to guess the factors when deriving relations, especially if the quantity depends on multiple variables.
It is not applicable when the expression involves more than three independent variables or when the quantity includes more than one term with different dimensions (sum/difference).

NCERT Physics Part 1, Appendix A9 (Dimension Analysis) for additional dimensional tables and examples.
Conceptual and applied texts listed in the course material for deeper practice and variations of problems.