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) \[{{G}^{1/2}}{{h}^{1/2}}{{c}^{1/2}}\]
Correct Answer: A
Solution :
Time\[\propto {{c}^{x}}{{G}^{y}}{{h}^{z}}\Rightarrow T=k{{c}^{x}}{{G}^{y}}{{h}^{z}}\] Putting the dimensions in the above relation \[\Rightarrow \] \[[{{M}^{0}}{{L}^{0}}{{T}^{1}}]={{[L{{T}^{-1}}]}^{x}}{{[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]}^{y}}{{[M{{L}^{2}}{{T}^{-1}}]}^{z}}\] \[\Rightarrow \] \[[{{M}^{0}}{{L}^{0}}{{T}^{1}}]=[{{M}^{-y+z}}{{L}^{x+3y+2z}}{{T}^{-x-2y-z}}]\] 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 \[[{{G}^{1/2}}{{h}^{1/2}}{{c}^{-5/2}}]\]
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