Question

If energy $(E)$, velocity $(V)$ and time $(T)$ are chosen as the fundamental quantities, the dimensional formula of surface tension will be

• A. $[E{V}^{-2}{T}^{-2}]$
• B. $[E^{-2}{V}^{-1}{T}^{-3}]$
• C. $[E{V}^{-2}{T}^{-1}]$
• D. $[E{V}^{-1}{T}^{-2}]$

Answer 1

Answer: (A)

Let $S=k{E}^a{V}^b{T}^c$
where $k$ is a dimensionless constant.
Writing the dimensions on both sides, we get
$\left[M^1{L}^0{T}^{-2}\right]={\left[M{L}^2{T}^{-2}\right]}^a{\left[L{T}^{-1}\right]}^b[T{]}^c$
$=\left[M^a{L}^{2a+b}{T}^{-2a-b+c}\right]$
Applying principle of homogeneity of dimensions,
we get, $a=1\dots (i)$
$2a+b=0\dots (ii)$
$-2a-b+c=-2\dots (iii)$
Adding $(ii)$ and $(iii)$, we get
$c=-2$
From $(ii),b=-2a=-2$
$\therefore S=kE{V}^{-2}{T}^{-2}$
or $[S]=\left[E{V}^{-2}{T}^{-2}\right]$