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The speed of light $(c)$, gravitational constant $(G)$ and planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be
${G^{1/2}}{h^{1/2}}{c^{ - 5/2}}$
${G^{ - 1/2}}{h^{1/2}}{c^{1/2}}$
${G^{1/2}}{h^{1/2}}{c^{ - 3/2}}$
${G^{1/2}}{h^{1/2}}{c^{1/2}}$
Solution
Let time, $T \propto c^{x} G^{y} h^{z}$
$\Rightarrow T=k c^{x} G^{y} h^{z}$
Taking dimensions on both sides $\left[M^{0} L^{0} T^{1}\right]=\left[L T^{-1}\right]^{x}\left[M^{-1} L^{3} T^{-2}\right]^{y}\left[M L^{2} T^{-1}\right]^{z}$
$i . e$
$\left[M^{0} L^{0} T^{1}\right]=\left[M^{-y+z} L^{x+3 y+2 z} T^{-x-2 y-z}\right]$
Equating power of $M, L,$ Ton both sides, we get
$-y+z=0 \quad \ldots(1)$
$x+3 y+2 z=0 \quad \ldots(2)$
$-x-2 y-z=1 \quad \ldots .(3)$
From $(1) \Rightarrow z=y$
Adding (2) and $(3) \Rightarrow y+z=1$
or $2 y=1 \quad[\text { From }]$
i.e, $y=\frac{1}{2}$
$\therefore z=y=\frac{1}{2}$
Putting these values in (2) we get $x+\frac{3}{2}+1=0$ or $x=\frac{-5}{2}$
Hence$,[T]=\left[G^{1 / 2} h^{1 / 2} c^{-5 / 2}\right]$
Similar Questions
Match List $I$ with List $II$ and select the correct answer using the codes given below the lists :
| List $I$ | List $II$ |
| $P.$ Boltzmann constant | $1.$ $\left[ ML ^2 T ^{-1}\right]$ |
| $Q.$ Coefficient of viscosity | $2.$ $\left[ ML ^{-1} T ^{-1}\right]$ |
| $R.$ Planck constant | $3.$ $\left[ MLT ^{-3} K ^{-1}\right]$ |
| $S.$ Thermal conductivity | $4.$ $\left[ ML ^2 T ^{-2} K ^{-1}\right]$ |
Codes: $ \quad \quad P \quad Q \quad R \quad S $
