JEE Advanced 2023 Paper 2, Physics Section 1 Q2: Units And Dimensions

JEE Advanced2023Paper 2Physics Section 1

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$ , Planck's constant $h$ and the speed of light $c$ , as $Y = c^{α} h^{β} G^{γ}$ . Which of the following is the correct option?

A
$α = 7, β = - 1, γ = - 2$
B
$α = - 7, β = - 1, γ = - 2$
C
$α = 7, β = - 1, γ = 2$
D
$α = - 7, β = 1, γ = - 2$

Solution

Write the dimensions of each quantity:

$[Y] = M L^{- 1} T^{- 2}$ , $[c] = L T^{- 1}$ , $[h] = M L^{2} T^{- 1}$ , $[G] = M^{- 1} L^{3} T^{- 2}$ .

Substituting into $Y = c^{α} h^{β} G^{γ}$ and matching powers:

$M$ : $1 = β - γ \dots (i)$

$L$ : $- 1 = α + 2 β + 3 γ \dots (ii)$

$T$ : $- 2 = - α - β - 2 γ \dots (iii)$

Adding (ii) and (iii) eliminates $α$ : $- 3 = β + γ$ . Combined with (i), $β - γ = 1$ , this gives $β = - 1$ and $γ = - 2$ .

Putting these back in (ii): $- 1 = α - 2 - 6 \Rightarrow α = 7$ .

$∴ α = 7, β = - 1, γ = - 2$ .

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JEE Advanced 2023 Paper 2, Physics Section 1 Q2: Units And Dimensions

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