Ncert Exercises & Solutions

If Planck's constant ($h$) and speed of light in a 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?

a.	mass of electron ($m_e$)
b.	universal gravitational constant ($G$)
c.	charge of electron ($e$)
d.	mass of proton ($m_p$)

1.	(a), (c), (d)
2.	(a), (b), (d)
3.	(a), (b), (c)
4.	(b), (c), (d)

(2) Hint: Use the concept of conversion of units.

Step 1: Find the dimensions of mass.

$We know that dimension of h = [h] = [{ML}^{2} T^{- 1}]$
$[C] = [{LT}^{- 1}] [m_{θ}] = M$
$[G] = [M^{- 1} L^{3} T^{- 2}]$
$[e] = [AT] [m_{p}] = [M]$
$[\frac{hc}{G}] = \frac{[{ML}^{2} T^{- 1}] [{LT}^{- 1}]}{[M^{- 1} L^{3} T^{- 2}]} = [M^{2}]$
$M = \sqrt{\frac{hc}{G}}$

Step 2: Find the dimensions of length.

$Similarly, \frac{h}{c} = \frac{[{ML}^{2} T^{- 1}]}{[{LT}^{- 1}]} = [ML]$
$L = \frac{h}{cM} = \frac{h}{c} \sqrt{\frac{G}{hc}} = \frac{\sqrt{Gh}}{c^{3 / 2}}$
$As, c = {LT}^{- 1}$
$\Rightarrow [T] = \frac{[L]}{[C]} = \frac{\sqrt{Gh}}{c^{3 / 2} \cdot C} = \frac{\sqrt{Gh}}{C^{5 / 2}}$

Hence, (a), (b), or (d) any can be used to express L, M, and T in terms of three chosen fundamental quantities.

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a.	mass of electron (\(m_e\))
b.	universal gravitational constant (\(G\))
c.	charge of electron (\(e\))
d.	mass of proton (\(m_p\))