Question

If the velocity of light c, gravitational constant G and Planck's constant h are chosen h as fundamental units, what are the dimensions of mass, length and time in the new system?

c = [L] [T]^(-1)
G = [M]^(-1) [L]^(3) [T]^(-2)
h = [M]^(1) [L]^(2)[T]^(-1)
Thanks!

Answer 1

`[m] = c^(1/2) G^(-1/2) h^(1/2)`

Explanation:

Here, `[c] = LT^-1`
`[G] = M^-1L^3T^-2`
`[h] = M^(1) L^(2)T^(-1)`
Let `m alpha c^xG^yh^z`

By substituting the dimensions of each quantity in both the sides,

`M = (LT^-1)^x (M^-1L^3T^-2)^y (ML^2T^-1)^z`
`M = [M^(y+z)L^(x+3y+2z)T^(x-2y-z)]`
By equating the power of M, L, T in both the sides:

`-y+z=1, x+3y+2z=0, -x-2y-z=0`

By soling the above equation ,
We get, `x= 1/2 , y=-1/2 , and z= 1/2`

So, `[m] = c^(1/2) G^(-1/2) h^(1/2)`

In above Manner, solve for Length And Time.