Which Of The Following Equations Are Dimensionally Consistent

Let's dive into the fascinating world of dimensional analysis and explore how we can determine if an equation is dimensionally consistent. This powerful technique allows us to verify the validity of equations in physics and engineering by ensuring that the dimensions on both sides of the equation match. An equation is dimensionally consistent if each term in the equation has the same dimensions. This principle is rooted in the idea that you can only add or subtract quantities that have the same physical units.

Dimensional Consistency: A Foundation of Physics

Dimensional analysis is a fundamental tool used to check the plausibility of equations and calculations in physics and engineering. It operates on the principle that every physical quantity has a dimension, which can be expressed in terms of basic dimensions such as mass (M), length (L), and time (T). Dimensional consistency requires that all terms in an equation have the same dimensions. In simpler terms, you can only add or subtract quantities that have the same units (e.g., you can add meters to meters, but you cannot add meters to seconds).

Why is dimensional consistency important?

• Error Detection: It helps to identify mistakes in derivations or calculations. If an equation is not dimensionally consistent, it's guaranteed to be wrong.
• Equation Validation: It provides a quick check of whether an equation could potentially be correct. If it passes the dimensional consistency test, it increases confidence in its validity.
• Understanding Physical Relationships: It reveals the relationships between different physical quantities.
• Unit Conversion: It aids in converting units from one system to another.
• Model Scaling: It is essential in scaling physical models and simulations.

Fundamental Dimensions and Derived Dimensions

Before we delve into specific equations, let's clarify the concept of dimensions. The most common fundamental dimensions are:

• Mass (M): Represents the amount of matter in an object.
• Length (L): Represents spatial extent.
• Time (T): Represents the duration of events.
• Electric Charge (Q): Represents the fundamental property of matter that causes it to experience a force in an electromagnetic field.
• Temperature (Θ): Represents the degree of hotness or coldness of a substance.

Other dimensions, known as derived dimensions, can be expressed in terms of these fundamental dimensions. Here are a few examples:

• Area: L² (Length squared)
• Volume: L³ (Length cubed)
• Velocity: L/T (Length per Time)
• Acceleration: L/T² (Length per Time squared)
• Force: M L/T² (Mass times Acceleration)
• Energy: M L²/T² (Mass times Velocity squared)
• Power: M L²/T³ (Energy per Time)
• Density: M/L³ (Mass per Volume)
• Pressure: M/(L T²) (Force per Area)

The Process of Checking Dimensional Consistency

The process of checking dimensional consistency involves the following steps:

1. Identify the equation you want to analyze.
2. Determine the dimensions of each term in the equation.
3. Substitute the dimensions of each term into the equation.
4. Simplify the equation by canceling out any common dimensions.
5. Verify that the dimensions on both sides of the equation are the same.

If the dimensions on both sides of the equation match, the equation is dimensionally consistent. If they do not match, the equation is dimensionally inconsistent and therefore incorrect.

Examples of Dimensional Consistency Checks

Let's examine several equations and determine their dimensional consistency:

Example 1: The Equation of Motion (Constant Acceleration)

Equation: d = v₀t + (1/2)at²

Where:

• d = displacement (length)
• v₀ = initial velocity (length/time)
• t = time
• a = acceleration (length/time²)

Dimensional Analysis:

• Dimension of d: L
• Dimension of v₀t: (L/T) * T = L
• Dimension of (1/2)at²: (L/T²) * T² = L

Since all terms have the dimension of length (L), the equation is dimensionally consistent.

Example 2: The Ideal Gas Law

Equation: PV = nRT

Where:

• P = pressure (force/area)
• V = volume
• n = number of moles (dimensionless)
• R = ideal gas constant (energy/mole/temperature)
• T = temperature

Dimensional Analysis:

• Dimension of P: M/(L T²)
• Dimension of V: L³
• Dimension of PV: [M/(L T²)] * L³ = M L²/T² (Energy)
• Dimension of n: 1 (dimensionless)
• Dimension of R: M L²/(T² Θ)
• Dimension of T: Θ
• Dimension of nRT: 1 * [M L²/(T² Θ)] * Θ = M L²/T² (Energy)

Since both sides of the equation have the dimension of energy (M L²/T²), the equation is dimensionally consistent.

Example 3: Einstein's Mass-Energy Equivalence

Equation: E = mc²

Where:

• E = energy
• m = mass
• c = speed of light (length/time)

Dimensional Analysis:

• Dimension of E: M L²/T²
• Dimension of m: M
• Dimension of c: L/T
• Dimension of c²: (L/T)² = L²/T²
• Dimension of mc²: M * (L²/T²) = M L²/T²

Since both sides of the equation have the dimension of energy (M L²/T²), the equation is dimensionally consistent.

Example 4: An Incorrect Equation (Hypothetical)

Equation: v = at² (Let's assume this is proposed as a velocity equation)

Where:

• v = velocity (length/time)
• a = acceleration (length/time²)
• t = time

Dimensional Analysis:

• Dimension of v: L/T
• Dimension of at²: (L/T²) * T² = L

The dimension of velocity (L/T) is not equal to the dimension of at² (L). Therefore, this equation is dimensionally inconsistent and is therefore incorrect. This quick check tells us that the relationship as written cannot be physically valid.

Example 5: A Simple Pendulum's Period

Equation: T = 2π√(L/g)

Where:

• T = period (time)
• L = length of the pendulum
• g = acceleration due to gravity (length/time²)
• 2π = dimensionless constant

Dimensional Analysis:

• Dimension of T: T
• Dimension of L: L
• Dimension of g: L/T²
• Dimension of L/g: L / (L/T²) = T²
• Dimension of √(L/g): √(T²) = T
• Dimension of 2π√(L/g): 1 * T = T

Since both sides of the equation have the dimension of time (T), the equation is dimensionally consistent.

Example 6: Navier-Stokes Equation (simplified term)

Let's consider just one term from the Navier-Stokes equation, a complex equation in fluid dynamics:

Term: ρ(v ⋅ ∇)v

Where:

• ρ = density (mass/volume)
• v = velocity (length/time)
• ∇ = gradient operator (1/length)

Dimensional Analysis:

• Dimension of ρ: M/L³
• Dimension of v: L/T
• Dimension of ∇: 1/L
• Dimension of (v ⋅ ∇): (L/T) * (1/L) = 1/T
• Dimension of (v ⋅ ∇)v: (1/T) * (L/T) = L/T²
• Dimension of ρ(v ⋅ ∇)v: (M/L³) * (L/T²) = M/(L²T²)

The resulting dimension, M/(L²T²), represents the dimensions of force per unit volume, which is consistent with the term's role in the equation representing forces acting on a fluid element. This specific term is dimensionally consistent within the larger Navier-Stokes equation. Checking each term in the full equation ensures the entire equation's validity.

Example 7: Bernoulli's Equation (simplified form)

Equation: P + (1/2)ρv² + ρgh = constant

Where:

• P = pressure (force/area)
• ρ = density (mass/volume)
• v = velocity (length/time)
• g = acceleration due to gravity (length/time²)
• h = height (length)

Dimensional Analysis:

• Dimension of P: M/(L T²)
• Dimension of (1/2)ρv²: (M/L³) * (L/T)² = M/(L T²)
• Dimension of ρgh: (M/L³) * (L/T²) * L = M/(L T²)

All terms have the same dimension of M/(L T²), which is the dimension of pressure or energy density. Therefore, the equation is dimensionally consistent.

Example 8: Wave Equation

Equation: ∂²y/∂t² = v² ∂²y/∂x²

Where:

• y = displacement (length)
• t = time
• x = position (length)
• v = velocity (length/time)

Dimensional Analysis:

• Dimension of ∂²y/∂t²: L/T²
• Dimension of ∂²y/∂x²: L/L² = 1/L (Note: second derivative reduces the powers of length)
• Dimension of v²: (L/T)² = L²/T²
• Dimension of v² ∂²y/∂x²: (L²/T²) * (1/L) = L/T²

Since both sides have the dimension L/T², the wave equation is dimensionally consistent.

Example 9: Radioactive Decay Law

Equation: N(t) = N₀e^-λt

Where:

• N(t) = number of radioactive nuclei at time t (dimensionless)
• N₀ = initial number of radioactive nuclei (dimensionless)
• λ = decay constant (1/time)
• t = time
• e = base of the natural logarithm (dimensionless)

Dimensional Analysis:

• Dimension of N(t): 1 (dimensionless)
• Dimension of N₀: 1 (dimensionless)
• Dimension of λ: 1/T
• Dimension of t: T
• Dimension of λt: (1/T) * T = 1 (dimensionless)
• Dimension of e^-λt: 1 (dimensionless, as the exponent must be dimensionless)
• Dimension of N₀e^-λt: 1 * 1 = 1 (dimensionless)

Since both sides are dimensionless, the equation is dimensionally consistent. Note that the exponent of any function (like the exponential) must always be dimensionless.

Example 10: Capacitance

Equation: C = Q/V

Where:

• C = capacitance
• Q = electric charge
• V = electric potential (voltage)

Dimensional Analysis:

• Dimension of Q: Q (fundamental dimension of electric charge)
• Dimension of V: M L²/(T³ Q) (Energy per unit charge)
• Dimension of C: Q / [M L²/(T³ Q)] = Q² T³/(M L²)

Therefore, the dimension of capacitance is Q² T³/(M L²). This illustrates how even for less immediately obvious physical quantities, dimensional analysis provides a clear understanding of the relationships between fundamental dimensions. This derived dimension can then be used in checking the dimensional consistency of other equations involving capacitance.

Limitations of Dimensional Analysis

While incredibly useful, dimensional analysis has limitations:

• Dimensionless Constants: It cannot determine the values of dimensionless constants (like π or 1/2). These constants must be determined through experiment or more detailed theoretical analysis.
• Functional Form: It cannot reveal the exact functional form of an equation. For example, it can tell you that velocity depends on acceleration and time, but it won't tell you if it's v = at or v = at².
• Incomplete Equations: It can only check for consistency; it cannot guarantee that an equation is completely correct. An equation can be dimensionally consistent but still be wrong due to other factors.
• Complex Systems: In very complex systems with many interacting variables, applying dimensional analysis can become challenging.

Common Mistakes to Avoid

• Forgetting to include all terms: Ensure you analyze every term in the equation.
• Incorrectly determining dimensions: Double-check the dimensions of each physical quantity. Refer to a reliable table of dimensions if needed.
• Ignoring dimensionless constants: While they don't affect dimensional consistency, be aware of their presence and that dimensional analysis cannot determine them.
• Mixing units: Ensure all quantities are expressed in a consistent system of units before performing the dimensional analysis.

Conclusion

Dimensional consistency is a powerful and essential tool for physicists and engineers. It provides a quick and easy way to check the plausibility of equations and calculations. By ensuring that all terms in an equation have the same dimensions, we can catch errors, validate equations, and gain a deeper understanding of the relationships between physical quantities. While dimensional analysis has limitations, its benefits far outweigh its drawbacks, making it an indispensable part of the scientific and engineering toolkit. Remember to always be mindful of dimensions when working with physical equations, and use dimensional analysis to verify your results. This practice will save you time and effort by helping you avoid mistakes and build confidence in your work.