Question

Of the following quantities, which one has dimensions different from the remaining three ?

• A. Energy per unit volume
• B. Force per unit area
• C. Product of voltage and charge per unit volume
• D. Angular momentum

Answer 1

Answer: (D)

Dimensions of energy $E=[M{L}^2{T}^{-2}]$
Dimensions of volume $v=[L^3]$
Dimensions of force $F=[ML{T}^{-2}]$
Dimensions of area $A=[L^2]$
Dimensions of voltage $V=[M{L}^2{T}^{-3}{A}^{-1}]$
Dimensions of charge $q=[AT]$
Dimensions of angular momentum $L=[M{L}^2{T}^{-1}]$
$\therefore$ Dimensions of $\frac{E}{v}=\frac{[M{L}^2{T}^{-2}]}{[L^3]}=[M{L}^{-1}{T}^{-2}]$
Dimensions of $\frac{F}{A}=\frac{[ML{T}^{-2}]}{[L^2]}=[M{L}^{-1}{T}^{-2}]$
Dimensions of $\frac{V_q}{v}=\frac{[M{L}^2{T}^{-3}{A}^{-1}][AT]}{[L^3]}=[M{L}^{-1}{T}^{-2}]$
Dimensions of angular momentum is $[M{L}^2{T}^{-1}]$
while other three has dimensions $[M{L}^{-1}{T}^{-2}]$