The speed of light ( c), gravitational constant (G) and plank's constant (h) are taken as fundamental units in a system. The dimensions of time in this new system should be.

Options

A. `G^(1//2) h^(1//2)c^(-5//2)`
B. `G^(-1//2) h^(1//2)c^(1//2)`
C. `G^(1//2) h^(1//2)c^(-3//2)`
D. `G^(1//2) h^(1//2)c^(1//2)`

Answer

a

Explanation

We have time `prop c^(x) G^(y) h^(x)`
`rArr T = kc^(x) G^(y) h^(z)`
According to the method of dimensional analysis the dimensional of each term on both sides of are equation must be same putting the dimensions formula in above relation
`rArr [M^(0)L^(0)T^(1)] = [LT^(-1)]^(x) [M^(-1)L^(3)T^(-2)]^(y)[ML^(2)T^(-1)]^(x)`
`rArr [M^(0)L^(0)T^(1)] =[M^(-y +z)L^(x+ 3y +2x) T^(-x -2y -x)]`
Comparing the dimensions of `M,L` and `T`
`-y + z = 0` ...(i)
`x + 3y + 2z = 0`....(ii)
`-x - 2y -z = 1`...(iii)
On solving Eqs `(i),(ii)` and `(iii)`
`x = - (5)/(2) , y = z = (1)/(2)`
Hence, dimensions of time are `[G^(1//2)h^(1//2)c^(-5//2)]`