A) \[\frac{\sigma }{\rho \lambda }\]
B) \[\frac{\rho }{\sigma \lambda }\]
C) \[\frac{\lambda }{\sigma \rho }\]
D) \[\rho \lambda \sigma \,\]
Correct Answer: A
Solution :
\[Let\,\,\,\nu \propto \,\,{{\sigma }^{a}}{{\rho }^{b}}{{\lambda }^{c}}\] Equating dimensions on both sides, \[\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]\propto {{\left[ {{M}^{T-2}} \right]}^{a}}\,{{\left[ M{{L}^{-3}} \right]}^{b}}{{\left[ L \right]}^{c}}\] Equating the powers of M, L, T on both sides, we get \[\operatorname{a}+b=0\] \[-\,3b+c=1\] Solving, we get \[a=\frac{1}{2},\,\,b=-\frac{1}{2},\,\,c=-\frac{1}{2}\] \[\therefore \,\,\,\,\,\nu \propto {{\sigma }^{1/2}}{{\rho }^{-1/2}}{{\lambda }^{-1/2}}\] \[\therefore \,\,\,\,\,\nu \propto \frac{\sigma }{\rho \lambda }\]
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