JEE Physics Units and Dimensions Notes Set 03

Dimensions of a Quantity

The dimensions of a physical quantity are the powers to which the base quantities (M, L, T, A, K) are raised to represent it. For example, force has dimensions \( [MLT^{-2}] \), energy \( [ML^2T^{-2}] \), pressure \( [ML^{-1}T^{-2}] \) and angular momentum \( [ML^2T^{-1}] \).

Principle of Homogeneity

Only quantities with the same dimensions can be added, subtracted or equated. Arguments of trigonometric, exponential and logarithmic functions are dimensionless.

Uses and Limits

1. Check the correctness of a relation. 2. Convert a quantity from one system of units to another: \( n_2=n_1\left[\frac{M_1}{M_2}\right]^a\left[\frac{L_1}{L_2}\right]^b\left[\frac{T_1}{T_2}\right]^c \). 3. Derive a relation among quantities by matching powers.
Limits: dimensionless constants cannot be found, and the method fails when a quantity depends on a sum of terms or on trigonometric functions.

Useful Dimensional Equalities

Planck constant \( h=E/\nu \) has \( [ML^2T^{-1}] \). The time constants L/R, RC and \( \sqrt{LC} \) are times. The speed \( 1/\sqrt{\mu_0\varepsilon_0} \) is a velocity. Work, energy and torque share the dimensions \( [ML^2T^{-2}] \).

Worked Example

Find the dimensions of a in \( \left(P+\frac a{V^2}\right)(V-b)=RT \).
Solution: \( a/V^2 \) has the dimensions of P, so \( [a]=[P][V^2]=ML^{5}T^{-2} \).