If the constant of gravitation `(G)` , Planck's constant `(h)` and the velocity of light `(c)` be chosen as fundamental units. The dimension of the radius of gyration is

Options

A. `h^(1//2)c^(-3//2) G^(1//2)`
B. `h^(1//2)c^(3//2) G^(1//2)`
C. `h^(1//2)c^(-3//2) G^(-1//2)`
D. `h^(-1//2)c^(-3//2) G^(1//2)`

Answer

A

Explanation

Radius of gyration `k prop h^(x) c^(y) G^(z)`
`[L] = [ML^2 T^(-1)]^(x) [LT^(-1)]^(y)[M^(-1)L^3T^(-2)]^z`
`[L] = [M^(x-z)L^(2x+y+3z)T(-x-y-2z)]`
`x-z=0 " "...(i)`
`2x + y + 3z = 1 " "...(ii)`
` -x - y - 2z = 0" "...(iii)`
By equation (i), (ii) and (iii)
`x = z = 1 // 2`
`y = -3 // 2`
Hence dimension of radius of gyration are `[h^(1//2) c^(-3//2) G^(1//2)]`