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If the velocity of light $C$, the gravitational constant $G$ and Planck's constant $h$ are chosen as the fundamental units, the dimension of density in the new system is
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Verified Answer
The correct answer is:
$C^5 G^{-2} h^{-1}$
Let $\rho=\mathrm{kC}^{\mathrm{a}} \mathrm{G}^{\mathrm{b}} \mathrm{h}^{\mathrm{c}}$
$\begin{aligned}
& \Rightarrow \quad[\rho]=[\mathrm{C}]^{\mathrm{a}}[\mathrm{G}]^{\mathrm{b}}[\mathrm{h}]^{\mathrm{c}} \\
& \Rightarrow \quad \mathrm{ML}^{-3}=\mathrm{L}^{\mathrm{a}} \mathrm{T}^{-\mathrm{a}} \mathrm{M}^{-\mathrm{b}} \mathrm{L}^{3 \mathrm{~b}} \mathrm{~T}^{-2 \mathrm{~b}} \mathrm{M}^{\mathrm{c}} \mathrm{L}^{2 \mathrm{c}} \mathrm{T}^{-\mathrm{c}} \\
& \Rightarrow \quad 1=-\mathrm{b}+\mathrm{c},-3=\mathrm{a}+3 \mathrm{~b}+2 \mathrm{c}, 0=-\mathrm{a}-2 \mathrm{~b}-\mathrm{c} \\
& \Rightarrow \quad \mathrm{a}=5, \mathrm{~b}=-2 \text { and } \mathrm{c}=-1
\end{aligned}$
$\begin{aligned}
& \Rightarrow \quad[\rho]=[\mathrm{C}]^{\mathrm{a}}[\mathrm{G}]^{\mathrm{b}}[\mathrm{h}]^{\mathrm{c}} \\
& \Rightarrow \quad \mathrm{ML}^{-3}=\mathrm{L}^{\mathrm{a}} \mathrm{T}^{-\mathrm{a}} \mathrm{M}^{-\mathrm{b}} \mathrm{L}^{3 \mathrm{~b}} \mathrm{~T}^{-2 \mathrm{~b}} \mathrm{M}^{\mathrm{c}} \mathrm{L}^{2 \mathrm{c}} \mathrm{T}^{-\mathrm{c}} \\
& \Rightarrow \quad 1=-\mathrm{b}+\mathrm{c},-3=\mathrm{a}+3 \mathrm{~b}+2 \mathrm{c}, 0=-\mathrm{a}-2 \mathrm{~b}-\mathrm{c} \\
& \Rightarrow \quad \mathrm{a}=5, \mathrm{~b}=-2 \text { and } \mathrm{c}=-1
\end{aligned}$
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