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 $$Y = c^\alpha h^\beta G^\gamma$$. Which of the following is the correct option?

The dimensional formulae of the physical quantities are:

• Young’s modulus (stress): $$Y \rightarrow M^{1}L^{-1}T^{-2}$$
• Speed of light: $$c \rightarrow L^{1}T^{-1}$$
• Planck’s constant: $$h \rightarrow M^{1}L^{2}T^{-1}$$
• Gravitational constant: $$G \rightarrow M^{-1}L^{3}T^{-2}$$

Assume $$Y = c^{\alpha}h^{\beta}G^{\gamma}$$. Equate the dimensions on both sides.

Step 1: Write combined dimensions on the right-hand side.
$$ \begin{aligned} c^{\alpha}&:&& L^{\alpha}T^{-\alpha}\\ h^{\beta}&:&& M^{\beta}L^{2\beta}T^{-\beta}\\ G^{\gamma}&:&& M^{-\gamma}L^{3\gamma}T^{-2\gamma} \end{aligned} $$ Multiplying, $$ M^{\,\beta-\gamma}\;L^{\,\alpha+2\beta+3\gamma}\;T^{-\alpha-\beta-2\gamma} $$

Step 2: Equate powers of $$M,\,L,\,T$$ with those of $$Y$$.

$$ \begin{aligned} \text{M:}&\quad \beta-\gamma &=&\; 1\; \quad -(1)\\ \text{L:}&\quad \alpha+2\beta+3\gamma &=&\; -1\quad -(2)\\ \text{T:}&\quad -\alpha-\beta-2\gamma &=&\; -2\quad -(3) \end{aligned} $$

Step 3: Solve the simultaneous equations.

From $$(1):\; \beta = 1+\gamma$$.

Substitute in $$(3):\; -\alpha-(1+\gamma)-2\gamma = -2$$
$$\Rightarrow -\alpha-1-3\gamma = -2$$
$$\Rightarrow \alpha + 3\gamma = 1 \quad -(4)$$

Substitute $$\beta = 1+\gamma$$ in $$(2):$$
$$\alpha + 2(1+\gamma)+3\gamma = -1$$
$$\Rightarrow \alpha + 2 + 5\gamma = -1$$
$$\Rightarrow \alpha + 5\gamma = -3 \quad -(5)$$

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

Insert $$\gamma = -2$$ into $$(4):$$
$$\alpha + 3(-2) = 1 \;\Rightarrow\; \alpha = 7$$

Insert $$\gamma = -2$$ into $$\beta = 1+\gamma$$:
$$\beta = 1 - 2 = -1$$

Step 4: State the exponents.
$$\alpha = 7,\; \beta = -1,\; \gamma = -2$$

Hence, the correct choice is:
Option A which is: $$\alpha = 7,\; \beta = -1,\; \gamma = -2$$