If momentum (p), area (A) and time (T) are taken to be fundamental quantities, then energy has the dimensional formula.

Options

A. `(pA^(-1)T^(1)]`
B. `[p^(2)AT]`
C. `[pA^(-(1)/(2))T]`
D. `[pA^((1)/(2)T)]`

Explanation

Given, fundamental quantities are momentum (p), area (A) and time (T).
We can write energy E as
`E prop p^(A)A^(b)T^(c)`
`E= kp^(a)A^(A)T^(c)`
`:. [E]=[p]^(a)[A]^(A)[T]^(c)`
where k is dimensionless constant of proportionality
Dimensions of `E=[E]=[M^(1)L^(2)T^(-2)] and p]=[M^(1)L^(1)T^(-1)]`
`[A]=[L^(2)]`
`[T]=[T^(1)]`
Putting all the dimensions, we get
`M^(1)L^(2)T^(-2)=[M^(1)L^(1)T^(-1)]^(a)[L^(2)][T]^(c)`
`:. M^(1)L^(2)T^(-2)=M^(a)L^(2b+b)T^(-a+c)`
By comparison of power
`a=1, 2b+a=2`
`:.2b+1=2`
`b=(1)/(2)`
`-a+c=-2`
`c=-2+a=-2+1=-1`
Thus `|E|=[pA^((1)/(2))T^(-1)]`