First slide

Dimensions of physical quantities

Question

The speed of light (c), gravitational constant (G) and Planck's constant (h) are taken as the fundamental units in a system. The dimension of time in this system is:

Moderate

A

$G^{1 / 2} h^{1 / 2} c^{- 5 / 2}$
B

$G^{- 1 / 2} h^{1 / 2} c^{1 / 2}$
C

$G^{1 / 2} h^{1 / 2} c^{- 3 / 2}$
D

$G^{1 / 2} h^{1 / 2} c^{1 / 2}$

Solution

$Time \propto c^{x} G^{y} h^{z} \Rightarrow T = k c^{x} G^{y} h^{z}$
Putting the dimensions in the above relation
$\begin{array}{l} \Rightarrow [M^{0} L^{0} T^{1}] = {[L T^{- 1}]}^{x} {[M^{- 1} L^{3} T^{- 2}]}^{y} {[M L^{2} T^{- 1}]}^{z} \\ \Rightarrow [M^{0} L^{0} T^{1}] = [M^{- y + z} L^{x + 3 y + 2 z} T^{- x - 2 y - z}] \end{array}$
Comparing the powers of M, L and T
$\begin{array}{l} - y + z = 0 . . . . . . . . . . . (i) \\ x + 3 y + 2 z = 0 . . . . . . . . . . . (i i) \\ - x - 2 y - z = 1 . . . . . . . . . . (i i i) \end{array}$
On solving equations (i) and (ii) and (iii)
$x = \frac{- 5}{2}, y = z = \frac{1}{2}$
Hence dimension of time are $[G^{1 / 2} h^{1 / 2} c^{- 5 / 2}]$

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