Physics And Measurement Question 264
Question: A physical quantity of the dimensions of length that can be formed out of c, G and $ \frac{e^{2}}{4\pi {\varepsilon _{0}}} $ is
[c is velocity of light, G is universal constant of gravitation and e is charge]
Options:
A) $ c^{2}[ G\frac{e^{2}}{4\pi {\varepsilon _{0}}} ] $
B) $ \frac{1}{c^{2}}{{[ \frac{e^{2}}{G4\pi {\varepsilon _{0}}} ]}^{1/2}} $
C) $ \frac{1}{c}G\frac{e^{2}}{4\pi {\varepsilon _{0}}} $
D) $ \frac{1}{c^{2}}{{[ G\frac{e^{2}}{4\pi {\varepsilon _{0}}} ]}^{1/2}} $
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Answer:
Correct Answer: D
Solution:
[d] Let dimensions of length is related as, $ L={{[c]}^{x}}{{[G]}^{y}}{{[ \frac{e^{2}}{4\pi {\varepsilon _{0}}} ]}^{z}}\Rightarrow \frac{e^{2}}{4\pi {\varepsilon _{0}}}=ML^{3}{{T}^{-2}} $
$ L={{[L{{T}^{-1}}]}^{x}}{{[{{M}^{-1}}L^{3}{{T}^{-2}}]}^{y}}{{[ML^{3}{{T}^{-2}}]}^{z}} $
$ [L]=[{{L}^{x+3y+3z}}{{M}^{-y+z}}{{T}^{-x-2y-2z}}] $
Comparing both sides $ z=y=\frac{1}{2},x=-2 $
Hence, L= $ {{c}^{-2}}{{[ G.\frac{e^{2}}{4\pi {\varepsilon _{0}}} ]}^{1/2}} $
