Animated Solution for Physics - Physics and Measurement: 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 ?

Imagine you are a physicist trying to understand how different fundamental constants relate to macroscopic properties of materials. In this problem, we are given a fascinating relationship from the JEE Advanced 2023 paper. We are told that Young's modulus of elasticity, denoted by $Y$, can be expressed in terms of three fundamental constants of nature: the speed of light $c$, Planck's constant $h$, and the universal gravitational constant $G$.

The equation provided is $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. This equation acts as a bridge between the macroscopic world (represented by Young's modulus) and the fundamental constants of the universe. Our mission is to find the exact numerical values of these mysterious powers: $\alpha$, $\beta$, and $\gamma$.

To unlock the secrets of this equation, we must translate it into the language of dimensions. Every physical quantity can be expressed in terms of fundamental dimensions: Mass $[M]$, Length $[L]$, and Time $[T]$. Let's recall the dimensional formulas for all the physical quantities involved in our master equation.

Young's modulus $Y$ is defined as stress over strain. Since strain is dimensionless, $Y$ has the same dimensions as stress (or pressure), which is force per unit area. Therefore, its dimensions are $[M{L}^{-1}{T}^{-2}]$.

The speed of light $c$ is simply a velocity, so its dimensions are $[L{T}^{-1}]$. Planck's constant $h$ relates energy to frequency ($E = h u$), giving it dimensions of $[M{L}^2{T}^{-1}]$. Finally, the gravitational constant $G$ comes from Newton's law of universal gravitation ($F=G{m}_1{m}_2/r^2$), yielding dimensions of $[M^{-1}{L}^3{T}^{-2}]$.

Now comes the magic of dimensional analysis. The principle of dimensional homogeneity states that for any physically meaningful equation, the dimensions on the left-hand side must perfectly match the dimensions on the right-hand side. It is like balancing a cosmic scale!

We substitute our dimensional building blocks back into the original equation:

Next, we need to group all the powers of mass $M$, length $L$, and time $T$ together on the right side. By applying the fundamental laws of exponents, we multiply the powers inside each bracket with the power outside, and then add the exponents for terms with the same base. This simplifies the right side beautifully:

Since the left side must equal the right side, the power of $M$ on the left must be exactly equal to the power of $M$ on the right. The same strict rule applies to the powers of $L$ and $T$. Equating these powers gives us a system of three linear equations:

For $M$: $1=\beta -\gamma$

For $L$: $-1=\alpha +2\beta +3\gamma$

For $T$: $-2=-\alpha -\beta -2\gamma$

We have just transformed a complex physics problem into a simple mathematical puzzle. Let's solve this system. Look closely at the equations for $L$ and $T$. The $\alpha$ terms have opposite signs! If we simply add the $L$ equation and the $T$ equation together, the $\alpha$ terms will beautifully cancel each other out.

Adding them gives: $-3=\beta +\gamma$. We already know from the $M$ equation that $1=\beta -\gamma$. If we add these two new equations, the $\gamma$ terms cancel out, giving us $-2=2\beta$. Dividing by two, we find that $\beta =-1$.

Substituting this back into $1=\beta -\gamma$, we easily find that $\gamma =-2$.

We are in the final stretch now! We have successfully found the values of $\beta$ and $\gamma$. All that is left is to find $\alpha$. Let's take our values and substitute them back into our equation for $L$:

This simplifies to $-1=\alpha -2-6$, which means $-1=\alpha -8$. Moving the $-8$ to the other side, we get $\alpha =7$.

The puzzle is complete. We have found all three powers: $\alpha =7$, $\beta =-1$, and $\gamma =-2$. This perfectly matches option (A). Dimensional analysis is truly a powerful tool, not just for checking equations, but for discovering the hidden relationships between the fundamental constants of our universe!