Animated Solution for Physics - Physics and Measurement: In form of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light), the time period will be proportional to

Imagine trying to build the universe from scratch. You would need some fundamental building blocks, right? In physics, we have these universal constants that dictate the rules of the cosmos. We have $G$, the universal gravitational constant, which tells us how strongly masses attract each other. We have $h$, Planck's constant, which sets the scale for the quantum world. And we have $c$, the speed of light, the ultimate cosmic speed limit.

But what if we wanted to construct a fundamental unit of time using only these three constants? This isn't just a mathematical exercise; it's a profound journey into the very fabric of reality. To do this, we turn to one of the most powerful tools in a physicist's arsenal: Dimensional Analysis.

We start by assuming that this fundamental time period, $t$, is proportional to some combination of $G$, $h$, and $c$. We can express this mathematically by raising each constant to an unknown power:

Our mission is to find the exact values of $a$, $b$, and $c$.

To solve this puzzle, we need to break down each of these constants into their fundamental dimensions: Mass ($M$), Length ($L$), and Time ($T$).

Let's start with time itself. The dimension of time is simply:

Next, we need the dimensions of our universal constants. If you ever forget them, you can quickly derive them from familiar formulas. For $G$, we use Newton's law of gravitation ($F=\frac{G{m}_1{m}_2}{r^2}$), which gives us:

For Planck's constant $h$, we can use the energy of a photon ($E = h u$), leading to:

And for the speed of light $c$, it's simply a velocity:

Now, we invoke the Principle of Dimensional Homogeneity. This principle states that for any valid physical equation, the dimensions on the left-hand side must perfectly match the dimensions on the right-hand side.

Let's substitute our dimensional formulas into our initial assumed relationship:

To make sense of the right side, we need to combine the powers of $M$, $L$, and $T$. When we multiply terms with the same base, we add their exponents. Let's do this carefully:

For $M$: The power becomes $-a+b$. For $L$: The power becomes $3a+2b+c$. For $T$: The power becomes $-2a-b-c$.

Putting it all together, our equation becomes:

We have now transformed a physics problem into a system of linear equations. By equating the powers of $M$, $L$, and $T$ on both sides, we can solve for our unknowns.

Step 1: Equating the powers of Mass ($M$) On the left, the power is $0$. On the right, it is $-a+b$. This is a great start! We now know that $G$ and $h$ must be raised to the exact same power.

Step 2: Equating the powers of Length ($L$) On the left, the power is $0$. On the right, it is $3a+2b+c$. Since we just found that $a=b$, we can substitute $a$ for $b$ in this equation: Now we have expressed $c$ in terms of $a$ as well.

Step 3: Equating the powers of Time ($T$) On the left, the power is $1$. On the right, it is $-2a-b-c$. Let's substitute $b=a$ and $c=-5a$ into this equation:

With $a$ found, the rest of the dominoes fall instantly. Since $b=a$, we have $b=\frac{1}{2}$. And since $c=-5a$, we get $c=-\frac{5}{2}$.

Planck Time

We have successfully found our exponents! Let's substitute them back into our original proportionality:

We can rewrite these fractional exponents using square roots and fractions to make it look much more elegant:

This matches option (a) perfectly. But this isn't just an answer to a multiple-choice question. This specific combination of constants is known as the Planck Time.

It is approximately $5.39\times 10^{-44}$ seconds. In the realm of physics, this is considered the smallest meaningful unit of time. It is the time it takes for light to travel one Planck length. Before this time interval (like in the very first moments of the Big Bang), our current laws of physics, including general relativity and quantum mechanics, break down and cease to make sense. You haven't just solved a math problem; you've derived the fundamental tick-rate of the universe!