Animated Solution for Physics - Physics and Measurement: If $e$ is the electronic charge, $c$ is the speed of light in free space and $h$ is Planck's constant, the quantity $\frac{1}{4\pi {\epsilon }_0}\frac{|e{|}^2}{hc}$ has dimensions of

Welcome, future engineers and physicists! Today, we are going to tackle a beautiful problem from dimensional analysis. At first glance, this question looks like a tedious exercise in algebra. It asks us to find the dimensions of a rather intimidating expression:

If you are like most students, your first instinct might be to roll up your sleeves and start finding the dimensional formula for each individual constant. You might recall that the charge $e$ has dimensions of $[IT]$, the speed of light $c$ is $[L{T}^{-1}]$, and Planck's constant $h$ is $[M{L}^2{T}^{-1}]$.

Then comes the tricky part: the permittivity of free space, ${\epsilon }_0$. Finding its dimensions requires recalling Coulomb's law, isolating ${\epsilon }_0$, and carefully managing a flurry of negative exponents. While this "brute force" method is perfectly valid and will eventually lead you to the correct answer, it is a minefield for silly mistakes. In the high-pressure environment of the JEE, time and accuracy are your most valuable assets.

So, let's take a step back. Let's look at this expression not as a random collection of letters, but as a symphony of physical laws.

Grouping by Physics

As a JEE aspirant, you must train your mind to recognize patterns. When you see ${\epsilon }_0$ and a charge squared ($e^2$) in the same expression, an alarm bell should ring in your head: Coulomb's Law!

Similarly, when you see Planck's constant $h$ multiplied by the speed of light $c$, another alarm should ring: Photon Energy!

Let's strategically rearrange our expression to isolate these familiar physical groups:

By grouping the terms this way, we have transformed an abstract dimensional analysis problem into a conceptual physics problem. Let's decode each group one by one.

Coulomb's Law

Let's focus on the first group: $\frac{e^2}{4\pi {\epsilon }_0}$.

Recall the electrostatic force between two point charges. If we have two electrons separated by a distance $r$, Coulomb's law tells us that the force $F$ between them is:

Now, let's cross-multiply by $r^2$ to isolate our target group:

This is a massive breakthrough! We don't need to know the messy dimensions of ${\epsilon }_0$. We just need to know the dimensions of Force and distance squared.

The dimensional formula for force is $[ML{T}^{-2}]$, and for distance squared, it is $[L^2]$. Multiplying them together is a breeze:

Just like that, we have conquered the numerator. No complex fractions, no risk of messing up the powers of current or time.

The Quantum Connection

Now, let's turn our attention to the denominator: $hc$.

Where do we encounter the product of Planck's constant and the speed of light? In the realm of modern physics! The energy $E$ of a photon is inversely proportional to its wavelength $\lambda$, and the equation is:

Once again, let's cross-multiply to isolate our target group:

This tells us that the dimensions of $hc$ are simply the dimensions of Energy multiplied by the dimensions of length.

We know that Energy (which is equivalent to work done) has the dimensional formula $[M{L}^2{T}^{-2}]$. The wavelength $\lambda$ is just a length, so its dimension is $[L]$. Let's multiply them:

We have successfully decoded both parts of our expression using pure physical intuition. Let's bring them back together.

We found that the numerator group has dimensions of $[M{L}^3{T}^{-2}]$, and the denominator group also has dimensions of $[M{L}^3{T}^{-2}]$.

Let's substitute these back into our original rearranged expression:

Look at that beauty! The numerator and the denominator are exactly identical. When we divide them, every single fundamental dimension—Mass, Length, and Time—cancels out perfectly.

The result is a pure, dimensionless number.

In physics, this specific combination of constants is not just a random textbook problem. It is known as the fine-structure constant, usually denoted by the Greek letter $\alpha$.

Because it is dimensionless, its value is the same no matter what system of units you use—whether it's SI units, CGS units, or even an alien civilization's system of measurement! Its value is approximately $\frac{1}{137}$.

The fine-structure constant is one of the most fundamental numbers in the universe. It characterizes the strength of the electromagnetic interaction between elementary charged particles. It dictates how tightly electrons are bound to the nucleus and determines the fine details of atomic spectra.

By solving this problem elegantly, you didn't just save time; you connected classical electromagnetism with quantum mechanics, just like the great physicists of the 20th century. Keep looking for these hidden connections, and physics will become a beautiful story rather than a list of formulas!