Which of the following product of \(e, h, \mu, G\) (where \(\mu\) is the permeability) be taken so that the…

Which of the following product of \(e, h, \mu, G\) (where \(\mu\) is the permeability) be taken so that the dimensions of the product are same as that of speed of light?
  1. \(h e^{-2} \mu^{-1} G^{0}\)
  2. \(h^{2} e G^{0} \mu\)
  3. \(h^{0} e^{2} G^{-1} \mu\)
  4. \(h G e^{-2} \mu^{0}\)

Solution

Here \(v=e^{a} h^{b} \mu^{c} G^{d}\). Taking the dimensions
\(\begin{aligned}
M^{0} L T^{-1} A^{0}=&\left[A T^{l}\right]^{a}\left[M L^{2} T^{-1}\right]^{b}\left[M L T^{-2} A^{-2}\right]^{c} \\
&\left[M^{-1} L^{3} T^{-2}\right]^{d}
\end{aligned}\)
There will be 4 simultaneous equations by equating the dimensions of \(M, L, T\) and \(A\). These are \(a-2 c=0\)
\(a-b-2 c-2 d=-1, b+c-d=0\)
and \(2 b+c+\) \(3 d=1\)
Solving for ' \(a\) ',\(b^{\prime}\), ' \(c\) ' and ' \(d\) ' we get
\(a=-2, b=1, c=-1, d=0\)
Thus, \(v=e^{-2} h \mu^{-1} G^{0}\) ,

Asked in: JEE Mains - Units and Dimensions - Chapter Test

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