The dimensional formula of $\sigma_s \cdot k_B^{-1} \cdot b$ (where $\sigma_s$ is Stefan\'s constant, $k_B$ is Boltzmann constant, and $b$ is Wien\'s displacement constant) is:

Dimension: nature of a physical quantity expressed in terms of fundamental dimensions M (mass), L (length), T (time), I (current), K (temperature), N (amount), J (luminous intensity). Principle of dimensional homogeneity: in any valid physical equation, all terms must have the same dimensions. Uses: checking correctness of equations, deriving relations, unit conversion. Key: dimensional formula tells the type of quantity but not its numerical value or significance.

$c$ (speed of light): $LT^{-1}$. $h$ (Planck): $ML^2T^{-1}$. $k_B$ (Boltzmann): $ML^2T^{-2}K^{-1}$. $G$ (Gravitational): $M^{-1}L^3T^{-2}$. $\varepsilon_0$ (permittivity): $M^{-1}L^{-3}T^4I^2$. $\mu_0$ (permeability): $MLT^{-2}I^{-2}$. $R$ (gas constant): $ML^2T^{-2}K^{-1}N^{-1}$. $\sigma_s$ (Stefan): $MT^{-3}K^{-4}$. $b$ (Wien): $LK$.

Stefan-Boltzmann law: total power radiated per unit area $= \sigma_s T^4$; $\sigma_s = 5.67\times10^{-8}$ W m$^{-2}$ K$^{-4}$. Wien\'s displacement law: $\lambda_{max} T = b = 2.898\times10^{-3}$ m K. Peak wavelength shifts to shorter $\lambda$ as $T$ increases. Planck\'s radiation law: gives complete spectral distribution $B_\lambda(T)$; reduces to Wien\'s law for short wavelengths and Rayleigh-Jeans for long wavelengths.

Cannot determine: dimensionless constants (e.g., $2\pi$ in $T = 2\pi\sqrt{l/g}$), equations involving trigonometric functions (which must be dimensionless), addition or subtraction of quantities (only checks if dimensions are same). Cannot distinguish between dimensionally identical quantities (e.g., work and torque both $= ML^2T^{-2}$). Power of dimensional analysis: can derive the form of physical laws without knowing constants (e.g., $T \propto \sqrt{l/g}$ for pendulum).

Dimensional analysis checks whether an equation is possible, never whether it is correct. It cannot determine dimensionless constants, so a derivation gives $T \propto \sqrt{L/g}$ but never the factor $2\pi$. It also fails for equations containing sums of terms with different physical meanings, and for functions such as $\sin\theta$, $e^x$ and $\log x$, whose arguments must always be dimensionless — which is itself a useful check, because it tells you immediately that the quantity inside a logarithm is a ratio.

The rule differs by operation. In addition and subtraction the answer carries as many decimal places as the least precise term. In multiplication and division it carries as many significant figures as the least precise factor. Rounding is done once, at the end, never at intermediate steps, because repeated rounding accumulates error. Exact counted numbers and defined constants have infinite significant figures and never limit the answer.

Systematic errors shift every reading the same way — a zero error in a vernier caliper, or an instrument that has drifted out of calibration — and they can be removed once identified. Random errors scatter readings around the true value and are reduced by repeating the measurement, since the mean of n readings has an uncertainty smaller by a factor of $\sqrt{n}$. When quantities combine, absolute errors add in addition and subtraction, while relative errors add in multiplication and division. A power multiplies its own relative error by that power, so in $g = 4\pi^2L/T^2$ the timing error counts twice over — which is why the pendulum experiment always times many oscillations rather than one.