Basic Maths and Units and DimensionsMedium

Question

Dimensions of universal gravitational constant (G) in terms of Planck's constant

( $h$ ), distance ( $L$ ), mass ( $M$ ) and time ( $T$ ) are $\_\_\_\_$ .

Options

A.$\left\lbrack {hTLM}^{- 2} \right\rbrack$
B.$\left\lbrack {hT}^{- 1}{LM}^{- 2} \right\rbrack$
C.$\left\lbrack h{TL}^{2}M^{- 2} \right\rbrack$
D.$\left\lbrack h^{- 1}{\text{ }T}^{- 1}{LM}^{- 2} \right\rbrack$

Solution

Sol. [Key Concept: Express $G$ from energy $= GMm/r$ and $E = h/T$; balance the dimensional powers.]

Energy $= \frac{hc}{\lambda} = \frac{{GM}_{1}M_{2}}{r}$

$$\begin{matrix} & \left\lbrack {ML}^{2}{\text{ }T}^{- 2} \right\rbrack = \lbrack h\rbrack\left\lbrack T^{- 1} \right\rbrack = \lbrack G\rbrack\left\lbrack M^{2}{\text{ }L}^{- 1} \right\rbrack \\ & \lbrack G\rbrack = \left\lbrack M^{- 2}{\text{ }L}^{+ 1}{\text{ }T}^{- 1}{\text{ }h}^{+ 1} \right\rbrack \end{matrix}$$

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