Dimensions of Physical Constants
The dimensions of a physical constant are determined by algebraically isolating it in a dimensionally homogeneous defining equation; only constants multiplying like-dimensional quantities without conversion are dimensionless.
Why this shows up in the exam
Checking fundamental-constant tables · Constructing natural scales · Testing proposed physical laws
Learn the idea
A constant may be dimensionful when it connects quantities with different dimensional structure. A constant is not automatically a pure number; G, h, and epsilon₀ carry exactly the dimensions needed to balance their defining laws.
🧠 Memory hook: Isolate the constant; its dimensions are whatever the equation still needs.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- [G] = M⁻¹ L³ T⁻² — gravitational constant from Newton's law
- [h] = M L² T⁻¹ — Planck constant from E = h nu
- [k_B] = M L² T⁻² Theta⁻¹ — Boltzmann constant from thermal energy
How to approach it
- 1Write the defining law
- 2Isolate the constant symbolically
- 3Substitute dimensions and simplify
Common slip-ups that cost marks
- •Declaring every universal constant dimensionless
- •Using a memorised formula from the wrong unit system
- •Confusing numerical value with dimensional character
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A quantity Q is defined as Q = force^1 / speed^2. Which dimensional formula represents Q?
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