If the velocity of light 'c', Gravitational constant 'G' & Plank's constant 'h' be chosen as fundamental units, find the dimensions of mass, length & time in this new system .
Text Solution
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[M] = [h 1/2 . C 1/2 . G –1/2 ] ; [L] = [h 1/2 . C –3/2 . G 1/2 ] ; [T] = [h 1/2 . C –5/2 . G 1/2 ]
Sol. We have the equation
= F
= MLT –2
[G] = M –1 L 3 T –2 .......... (i)
= Energy Å
= ML 2 T –2 [c] = LT –1
[ λ ] = L
[h] = ML 2 T –1 .............. (ii)
[c] = LT –1 ................ (iii)
taking the product of (i) & (ii)
[G] [h] = L 5 T –3
[c] 3 = L 3 T –3
∴ 
= L 2
= L
again from (iii)
[T] = 
= 
= 
From (ii)
[h] = ML 2 T –1
[h] = 
[h] = MG 1/2 h 1/2 c –3+5/2
or G -1/2 h 1/2 c 1/2 = M
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