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

Let time `Tpropc^(X)G^(Y)h^(Z)`
`impliesT=kc^(X)G^(Y)h^(Z)`
Taking dimensions on both sides
`[M^(0)L^(0)T^(1)]`
`=[LT^(-1)]^(X)[M^(-1)L^(3)T^(-2)]^(Y)[ML^(2)T^(-1)]^(Z)`
i.e.
`[M^(0)L^(0)T^(1)]`
`=[M^(-Y+Z)L^(X+3Y+2Z)T^(-X-2Y-Z)]`
Equation powers of M,L and T on both sides we get
`-Y+Z=0`.........1
`X+3Y+2Z=0`.......2
`-X-2y-Z=1`..........3
From equation 1
`Y=Z`
Adding 2 and 3
`Y+Z=1`
or `2Y=1( :. Z=Y)`
`:.Z=Y=1/2`
Putting these values in (2) we get
`x+3/2+1=0` or `X=(-5)/2`
Hence `[T]=G^(1/2)h^(1/2)c^((-5)/2)]`