Question

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

• A. (a) Energy per unit volume
• A. (b) Force per unit area
• A. (c) Product of voltage and charge per unit volume
• A. (d) Angular momentum per unit mass

Answer 1

(d)

Sol. Energy per unit volume = $\frac{\lbrack ML^{2}T^{- 2}\rbrack}{\lbrack L^{3}\rbrack} = \lbrack ML^{- 1}T^{- 2}\rbrack$

Force per unit area = $\frac{\lbrack MLT^{- 2}\rbrack}{\lbrack L^{2}\rbrack} = \lbrack ML^{- 1}T^{- 2}\rbrack$

Product of voltage and charge per unit volume

$= \frac{V \times Q}{\text{Volume}} = \frac{VIt}{\text{Volume}} = \frac{\text{Power} \times \text{Time}}{\text{Volume}}$

$\Rightarrow$ $\frac{\lbrack ML^{2}T^{- 3}\rbrack\lbrack T\rbrack}{\lbrack L^{3}\rbrack} = \lbrack ML^{- 1}T^{- 2}\rbrack$

Angular momentum per unit mass = $\frac{\lbrack ML^{2}T^{- 1}\rbrack}{\lbrack M\rbrack} = \lbrack L^{2}T^{- 1}\rbrack$

So angular momentum per unit mass has different dimension.