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

Options

A. `[Ev^(-2)T^(-1)]`
B. `[Ev^(-1)T^(-2)]`
C. `[Ev^(-2)T^(-2)]`
D. `[E^(-2)v^(-1)T^(-3)]`

Answer

C

Explanation

We know that
Surface tension `(S) = ("Force"(F))/("Length"(L))`
So, `[S] = ([MLT^(-2)])/([L]) = [ML^(0)T^(-2)]`
Energy `(E) =` Force `xx` displacement
`rArr [E] = [ML^(2)T^(-2)]`
Velocity `(v) = ("displacement")/("time")`
`rArr [v] = [LT^(-1)]`
As, `S prop E^(a) v^(b) T^(c)`
where, a,b,c are constants From the principle of homogeneity,
`[LHS] = [RHS]`
`rArr [ML^(0)T^(-2)] = [ML^(2)T^(-2)]^(a) [LT^(-1)]^(b) [T]^(c)`
`rArr [ML^(0)T^(-2)] [M^(a)L^(2a+b) T^(-2a-b+c)]`
Equating the power on both sides, we get
`a =1, 2a +b = 0,b =- 2`
`rArr - 2a - b +c =- 2`
`rArr c = (2a +b) - 2 = 0 - 2 =- 2`
So `[S] = [Ev^(-2)T^(-2)] = [Ev^(-2)T^(-2)]`