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If Planck's constant $(h)$ and speed of light in vacuum $(c)$ are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities?
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Verified Answer
The correct answers are:
Mass of electron $\left(m_e\right)$
,
Universal gravitational constant $(G)$
,
Mass of proton $\left(m_p\right)$
Mass of electron $\left(m_e\right)$
,
Universal gravitational constant $(G)$
,
Mass of proton $\left(m_p\right)$
As we know that dimension of $[h]$
$$
=\frac{E}{v}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]
$$
Dimension of $[c]=\frac{s}{t}=\frac{[\mathrm{L}]}{[\mathrm{T}]}$ $=\left[\mathrm{LT}^{-1}\right]$
Dimension of mass of electron $\left[m_e\right]=\mathrm{M}$
$$
\begin{aligned}
&{[G]=\frac{F r^2}{M_1 M_2}=\frac{\left[\mathrm{ML}^3 \mathrm{~T}^{-2}\right]}{[\mathrm{M}][\mathrm{M}]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]} \\
&{[e]=[A T]}
\end{aligned}
$$
Dimension of mass of proton $\left[m_p\right]=[\mathbf{M}]$
$$
\begin{aligned}
&{\left[\frac{h c}{G}\right]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]\left[\mathrm{LT}^{-1}\right]}{\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^2\right]} \\
&\Rightarrow M=\sqrt{\frac{h c}{G}}
\end{aligned}
$$
Similarly,
$$
\frac{h}{c}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]}{\left[\mathrm{LT}^{-1}\right]}=[\mathrm{ML}]
$$
$$
\begin{aligned}
&L=\frac{h}{c M}=\frac{h}{c} \sqrt{\frac{G}{h c}}=\frac{\sqrt{G h}}{c^{3 / 2}} \\
&\text { As, } c=\left[\mathrm{LT}^{-1}\right] \\
&\Rightarrow[T]=\frac{[L]}{[c]}=\frac{\sqrt{G h}}{c^{3 / 2} \cdot c}=\frac{\sqrt{G h}}{c^{5 / 2}}
\end{aligned}
$$
Hence physical quantities (a), (b) or (d) any can be used to represent L, M and T in terms of three chosen fundamental quantities.
$$
=\frac{E}{v}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]
$$
Dimension of $[c]=\frac{s}{t}=\frac{[\mathrm{L}]}{[\mathrm{T}]}$ $=\left[\mathrm{LT}^{-1}\right]$
Dimension of mass of electron $\left[m_e\right]=\mathrm{M}$
$$
\begin{aligned}
&{[G]=\frac{F r^2}{M_1 M_2}=\frac{\left[\mathrm{ML}^3 \mathrm{~T}^{-2}\right]}{[\mathrm{M}][\mathrm{M}]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]} \\
&{[e]=[A T]}
\end{aligned}
$$
Dimension of mass of proton $\left[m_p\right]=[\mathbf{M}]$
$$
\begin{aligned}
&{\left[\frac{h c}{G}\right]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]\left[\mathrm{LT}^{-1}\right]}{\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^2\right]} \\
&\Rightarrow M=\sqrt{\frac{h c}{G}}
\end{aligned}
$$
Similarly,
$$
\frac{h}{c}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]}{\left[\mathrm{LT}^{-1}\right]}=[\mathrm{ML}]
$$
$$
\begin{aligned}
&L=\frac{h}{c M}=\frac{h}{c} \sqrt{\frac{G}{h c}}=\frac{\sqrt{G h}}{c^{3 / 2}} \\
&\text { As, } c=\left[\mathrm{LT}^{-1}\right] \\
&\Rightarrow[T]=\frac{[L]}{[c]}=\frac{\sqrt{G h}}{c^{3 / 2} \cdot c}=\frac{\sqrt{G h}}{c^{5 / 2}}
\end{aligned}
$$
Hence physical quantities (a), (b) or (d) any can be used to represent L, M and T in terms of three chosen fundamental quantities.
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