If energy (E), velocity (v) and time (T) are chosen as the fundamental quantities, the dimensional 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, `" "SpropE^(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)]`