Physics And Measurement Question 32
Question: The speed of light $ (c) $ , gravitational constant $ (G) $ and Planck’s constant $ (h) $ are taken as the fundamental units in a system. The dimension of time in this new system should be
[AMU 1995]
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}} $
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Answer:
Correct Answer: A
Solution:
Time $ \propto c^{x}G^{y}h^{z}\Rightarrow T=kc^{x}G^{y}h^{z} $
Putting the dimensions in the above relation
$ \Rightarrow $ $ [M^{0}L^{0}T^{1}]={{[L{{T}^{-1}}]}^{x}}{{[{{M}^{-1}}L^{3}{{T}^{-2}}]}^{y}}{{[ML^{2}{{T}^{-1}}]}^{z}} $
$ \Rightarrow $ $ [M^{0}L^{0}T^{1}]=[{{M}^{-y+z}}{{L}^{x+3y+2z}}{{T}^{-x-2y-z}}] $ Comparing the powers of $ M,L $ and $ T $
$ -y+z=0 $ -(i)
$ x+3y+2z=0 $ -(ii)
$ -x-2y-z=1 $ -(iii)
On solving equations (i) and (ii) and (iii) $ x=\frac{-5}{2},y=z=\frac{1}{2} $
Hence dimension of time are $ [{{G}^{1/2}}{{h}^{1/2}}{{c}^{-5/2}}] $
