A) \[[{{h}^{1}}{{c}^{1}}{{G}^{-1}}]\]
B) \[[{{h}^{1/2}}{{c}^{1/2}}{{G}^{-1/2}}]\]
C) \[[{{h}^{1}}{{c}^{-3}}{{G}^{1}}]\]
D) \[[{{h}^{1/2}}{{c}^{-3/2}}{{G}^{1/2}}]\]
Correct Answer: D
Solution :
Key Idea: Every equation relating physical quantities should be in dimensional balance. In order to establish relation among various physical quantities, let \[a,\,\,b,\,\,c\] be the powers to which \[h,\,\,c\] and \[G\] are raised, then \[[L]=[{{h}^{a}}{{c}^{b}}{{G}^{c}}]\] Putting the dimensions on \[RHS\] of above equation, we get \[[L]=[M{{L}^{2}}{{T}^{-1}}][L{{T}^{-1}}]{{[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]}^{c}}\] \[[L]=[{{M}^{a-c}}{{L}^{2a+b+3c}}{{T}^{-a-b-2c}}]\] Comparing the power, we get \[a-c=0\] ? (i) \[2a+b+3c=1\] ... (ii) \[-a-b-2c=0\] ... (iii) Solving Eqs. (i), (ii) and (iii), we get \[a=\frac{1}{2},\,\,b=\frac{-3}{2},\,\,c=\frac{1}{2}\] Hence,\[[L]=[{{h}^{1/2}}{{c}^{-3/2}}{{G}^{1/2}}]\]
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