Dimensional Method

Dimensional analysis is the cheapest sanity check in physics. It cannot give numerical constants, but it tells you instantly whether an answer can possibly be correct and lets you guess the form of unknown formulae from the relevant variables.

Concept

Each physical quantity carries dimensions in mass $M$ , length $L$ , time $T$ (and optionally $I$ , $\Theta$ ). Three uses dominate JEE:

Checking homogeneity. Every additive term must have identical dimensions.
Converting units. $n_1 [u_1] = n_2 [u_2]$ .
Deriving relations. If $y$ depends on $a,b,c$ , write $y = k\, a^x b^y c^z$ and match exponents.

Pure numbers, angles (radian) and arguments of $\sin, \log, \exp$ are dimensionless.

Derivation

Suppose the time period $T$ of a simple pendulum may depend on length $\ell$ , mass $m$ and gravity $g$ . Assume $T = k\,\ell^{a}\, m^{b}\, g^{c}.$ Matching dimensions $[T] = [L]^{a}[M]^{b}[L T^{-2}]^{c} = M^{b} L^{a+c} T^{-2c}$ :

$M$ : $b = 0$ ,
$T$ : $-2c = 1 \Rightarrow c = -1/2$ ,
$L$ : $a + c = 0 \Rightarrow a = 1/2$ .

So $T = k\sqrt{\ell/g}$ . The dimensionless constant $k = 2\pi$ requires another method.

JEE Worked Example

Q. The terminal velocity $v$ of a sphere through a viscous fluid depends on radius $r$ , viscosity $\eta$ and effective weight per unit volume $\rho g$ . Find $v$ by dimensions.

Solution. Let $v = k\, r^{a}\, \eta^{b}\, (\rho g)^{c}$ . Dimensions: $\eta = [M L^{-1} T^{-1}]$ , $\rho g = [M L^{-2} T^{-2}]$ .

$[L T^{-1}] = L^{a}\,(M L^{-1} T^{-1})^{b}\,(M L^{-2} T^{-2})^{c}.$

$M$ : $b + c = 0$ . $T$ : $-b - 2c = -1$ . Solving: $b = -1, c = 1$ . $L$ : $a - b - 2c = 1 \Rightarrow a = 2$ .

Therefore $v \propto r^{2}\rho g/\eta$ , matching Stokes' law with $k = 2/9$ .

Traps

Treating an exponent or trig argument as dimensional.
Assuming dimensional analysis gives the numerical constant.
Using it when a quantity depends on more than three independent dimensional groups (over-determined).
Forgetting that scalars and vectors of the same dimensions can still be physically inequivalent.

Key Takeaways

Match dimensions term-by-term; constants are dimensionless.
Three-variable power-law guesses are the bread and butter.
Dimensional methods fail for trigonometric/exponential dependencies and numerical prefactors.