First slide

Dimensions of physical quantities

Question

The dimension of stopping potential $V_{0}$ in photoelectric effect in units of Planck’s constant ‘h’, speed of light ‘C’ and universal gravitational constant ‘G’ and ampere A is:

Moderate

A

$h^{1 / 3} G^{2 / 3}, C^{1 / 3} A^{- 1}$
B

$h^{0}, C^{5}, G^{- 1}, A^{- 1}$
C

$h^{- 2 / 3}, C^{- 1 / 3}, G^{4 / 3}, A^{- 1}$
D

$h^{2}, G^{3 / 2}, C^{1 / 3}, A^{- 1}$

Solution

$V = K {(h)}^{a} {(A)}^{b} {(G)}^{c} {(C)}^{d}$ (V is voltage)
We know $[h] = M L^{2} T^{- 1}$
[A] = $A$
[G] = $M^{- 1} L^{3} T^{- 2}$
[C] = $L T^{- 1}$
[V] = ${ML}^{2} T^{-3} A^{- 1}$

comparing dimensions on both sides,
${ML}^{2} T^{-3} A^{- 1} = {(M L^{2} T^{- 1})}^{a} {(A)}^{b} {(M^{- 1} L^{3} T^{- 2})}^{c} {(L T^{- 1})}^{d}$
${⇒ML}^{2} T^{-3} A^{- 1} = M^{a - c} L^{2 a + 3 c + d} T^{- a - 2 c - d} A^{b}$
Comparing the powers,
$a - c = 1.................. (1)$
$b = - 1................. (2)$
$\begin{array}{l} - a - 2 c - d = - 3 ......... (3) \\ 2 a + 3 c + d = 2 ............. (4) \end{array}$
$\begin{array}{l} A d d 3 a n d 4, \\ a + c = - 1 ......... (5) \\ A d d 1 a n d 5, \\ 2 a = 0 \\ \Rightarrow a = 0 \\ ∴ c = - 1 \\ P u t v a l u e s o f a a n d c i n 3, \\ 2 - d = - 3 \\ \Rightarrow d = 5 \\ T h u s, V = K h^{0} A^{- 1} G^{- 1} C^{5} \end{array}$

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