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

Options

A. `(P^1 A^(-1) T^1)`
B. `(P^2 A^1 T^1)`
C. `(P^1 A^(1//2) T^1)`
D. `(P^1 A^(1//2)T^(-1))`

Answer

(d)

Explanation

Let energy, `U prop P^a A^b T^c`
`U = k P^a A^b T^c …..(i)`
where k is dimensionless constant of proportionality. Form (i) `[ML^2 T^(-2)] = (MLT^(-1))^a T^c = M^a L^(a+2b) T^(-a+c)`
Applying the principle of homogeneity of dimensions, we get
`a =1, a+2b = 2, b =(2-a)/(2) = (2-1)/(2) = (1)/(2)`
-a + c= -2, c= -2 + a = -2 +1 = -1
`:. from (i), U = k[P^1 A^(1//2) T^(-1)]`
Choice (d) is correct.