Revision/fast recall

1. TITLE

Cheat Sheet: Dimensions of High-Yield Constants — Complete revision in 3 minutes

2. QUICK FACTS BOX

DIMENSIONS & CONSTANTS — QUICK FACTS

  • Dimensions describe the exact nature of a physical quantity.
  • Base quantities are denoted by square brackets like $[L], [M], [T]$.
  • Some physical constants have dimensions, while pure numbers are dimensionless.
  • Dimensional analysis cannot find values of dimensionless constants.
  • Arguments of trig, log, and exponential functions are strictly dimensionless.
  • Initial velocity, speed, and average velocity all share the exact same dimension $[L T^{-1}]$.
  • Principle of Homogeneity states we can only add/subtract quantities with identical dimensions.

3. FORMULA SHEET

FORMULAS:

  • Gravitational Constant ($G$)
    → Use when: Extracting from unit $N~m^2~kg^{-2}$ to get $[M^{-1} L^3 T^{-2}]$.
  • Planck Constant ($h$)
    → Use when: Extracting from unit $J~s$ ($kg~m^2 s^{-1}$) to get $[M L^2 T^{-1}]$.
  • Boltzmann Constant ($k$)
    → Use when: Extracting from unit $J~K^{-1}$ ($kg~m^2 s^{-2} K^{-1}$) to get $[M L^2 T^{-2} K^{-1}]$.
  • Speed of light in vacuum ($c$)
    → Use when: Extracting from unit $m~s^{-1}$ to get $[L T^{-1}]$.

4. MUST-REMEMBER VALUES/CONSTANTS

KEY VALUES:

  • Planck constant ($h$) = $6.62607015 \times 10^{-34}~J~s$
  • Gravitational constant ($G$) = $6.67 \times 10^{-11}~N~m^2~kg^{-2}$
  • Boltzmann constant ($k$) = $1.380649 \times 10^{-23}~J~K^{-1}$
  • Speed of light ($c$) = $299792458~m~s^{-1}$
  • Avogadro constant ($N_A$) = $6.02214076 \times 10^{23}~mol^{-1}$

5. COMPARISON SUMMARY

Constant Dimensional Formula
Planck constant (h) $[M L^2 T^{-1}]$
Gravitational constant (G) $[M^{-1} L^3 T^{-2}]$
Boltzmann constant (k) $[M L^2 T^{-2} K^{-1}]$
Speed of light (c)  $[L T^{-1}]$

6. COMMON MISTAKES — QUICK LIST

DON'T:

  • Assume all physical constants are dimensionless; $G$, $h$, and $k$ have distinct dimensional formulae.
  • Forget that constants in the exponent (like $\lambda$ in $e^{-\lambda x}$) must make the entire exponent a dimensionless quantity.
  • Attempt to add or subtract constants or variables that do not share the exact same dimensions.

DO:

  • Derive the dimensional formula of a constant from its SI unit if you forget it during the exam.
  • Remember that dimensional correctness does not guarantee an equation is physically correct, due to unknown dimensionless constants.

7. MNEMONICS & MEMORY TRICKS

REMEMBER:
* Bracket Rule: Using square brackets $[~]$ always means you are dealing with the dimensions, not the magnitude.
* "Homogeneity Check": You can't add apples to oranges, and you can't add $[M]$ to $[L]$.

8. HIGH-YIELD NEET POINTS

NEET LOVES TO ASK:

  • Finding the dimensions of a constant embedded within an exponent or trig function (they must equate to $[M^0 L^0 T^0]$).
  • Deducing relationships using dimensional analysis, specifically equating the exponents of $[M]$, $[L]$, and $[T]$.
  • Identifying limitations of dimensional analysis: it fails to determine dimensionless constants like $\frac{1}{2}$ or $2\pi$.

9. ONE MCQ — SELF-TEST

Q: The SI unit of the Planck constant $h$ is given as $kg~m^2~s^{-1}$. What is its dimensional formula?
(a) $[M L T^{-1}]$
(b) $[M L^2 T^{-2}]$
(c) $[M L^2 T^{-1}]$
(d) $[M^2 L T^{-1}]$

Answer: (c) $[M L^2 T^{-1}]$ — Derived directly from its base units $kg$ ($[M]$), $m^2$ ($[L^2]$), and $s^{-1}$ ($[T^{-1}]$).

10. NCERT PAGE REFERENCES

Revise from:
- Page 2: Definitions and exact values of SI Base constants ($h$, $c$, $k$).
- Page 7-8: Dimensions of physical quantities and Principle of Homogeneity.
- Page 10: Exercises testing dimensions of Universal Gravitational Constant ($G$).

11. CTA