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^αh^βG^γ. Which of the following is the correct option?

The correct option is α = 7, β = −1, γ = −2.

To find the values of exponents α, β, and γ, we use dimensional analysis by expressing all quantities in terms of fundamental dimensions: Mass [M], Length [L], and Time [T].

First, let us determine the dimensional formula for each given quantity:
1. Young’s modulus of elasticity (Y):
Young’s modulus is defined as stress divided by strain. Since strain is dimensionless:
Dimension of Y = [Stress] = [Force] / [Area]
$\left[Y\right]=\frac{\left[\text{M}\text{L}{\text{T}}^{-2}\right]}{\left[{\text{L}}^2\right]}=\left[\text{M}{\text{L}}^{-1}{\text{T}}^{-2}\right]$

2. Speed of light (c):
Speed has dimensions of length per unit time:
$\left[c\right]=\left[\text{L}{\text{T}}^{-1}\right]$

3. Planck’s constant (h):
Planck’s constant is given by energy divided by frequency:
$\left[h\right]=\frac{\left[\text{Energy}\right]}{\left[\text{Frequency}\right]}=\frac{\left[\text{M}{\text{L}}^2{\text{T}}^{-2}\right]}{\left[{\text{T}}^{-1}\right]}=\left[\text{M}{\text{L}}^2{\text{T}}^{-1}\right]$

4. Gravitational constant (G):
From Newton's law of gravitation, G = (Force × distance²) / (mass²):
$\left[G\right]=\frac{\left[\text{M}\text{L}{\text{T}}^{-2}\right]·\left[{\text{L}}^2\right]}{\left[{\text{M}}^2\right]}=\left[{\text{M}}^{-1}{\text{L}}^3{\text{T}}^{-2}\right]$

Now, substitute these dimensions into the given relation:
$Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$

$\left[\text{M}{\text{L}}^{-1}{\text{T}}^{-2}\right]={\left[\text{L}{\text{T}}^{-1}\right]}^{\alpha }·{\left[\text{M}{\text{L}}^2{\text{T}}^{-1}\right]}^{\beta }·{\left[{\text{M}}^{-1}{\text{L}}^3{\text{T}}^{-2}\right]}^{\gamma }$

Group the exponents for each fundamental dimension M, L, and T:
$\left[\text{M}{\text{L}}^{-1}{\text{T}}^{-2}\right]=\left[{\text{M}}^{\beta -\gamma }{\text{L}}^{\alpha +2\beta +3\gamma }{\text{T}}^{-\alpha -\beta -2\gamma }\right]$

Equating the powers of M, L, and T on both sides gives three linear equations:
1) Power of M:
$\beta -\gamma =1$

2) Power of L:
$\alpha +2\beta +3\gamma =-1$

3) Power of T:
$-\alpha -\beta -2\gamma =-2$ ⇒ $\alpha +\beta +2\gamma =2$

Solving these equations step-by-step:
From equation (1), express β in terms of γ:
$\beta =1+\gamma$

Substitute $\beta =1+\gamma$ into equations (2) and (3):
Equation (2):
$\alpha +2(1+\gamma )+3\gamma =-1$
$\alpha +2+5\gamma =-1$ ⇒ $\alpha +5\gamma =-3$ --- (Equation 4)

Equation (3):
$\alpha +(1+\gamma )+2\gamma =2$
$\alpha +1+3\gamma =2$ ⇒ $\alpha +3\gamma =1$ --- (Equation 5)

Subtract Equation (5) from Equation (4):
$(\alpha +5\gamma )-(\alpha +3\gamma )=-3-1$
$2\gamma =-4$ ⇒ $\gamma =-2$

Now calculate β using $\beta =1+\gamma$:
$\beta =1+(-2)=-1$

Now calculate α using Equation (5):
$\alpha +3(-2)=1$
$\alpha -6=1$ ⇒ $\alpha =7$

Thus, the exponents are α = 7, β = −1, γ = −2.