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?

Question

c = [L] [T]^(-1)
G = [M]^(-1) [L]^(3) [T]^(-2)
h = [M]^(1) [L]^(2)[T]^(-1)
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Answer 1 · 2016-11-02T06:26:28

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

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.