Animated Solution for Physics - Physics and Measurement: A quantity $X$ is given by ${\epsilon }_{0}L\frac{\Delta V}{\Delta t}$, where ${\epsilon }_0$ is the permittivity of free space, $L$ is a length, $\Delta V$ is a potential difference and $\Delta t$ is a time interval. The dimensional formula for $X$ is the same as that of

When you first look at the expression $X={\epsilon }_{0}L\frac{\Delta V}{\Delta t}$, your initial instinct might be to panic. Finding the dimensional formula for the permittivity of free space (${\epsilon }_0$) from scratch using Coulomb's law is a tedious process. You would have to substitute the dimensions of force, charge, and distance, and then carefully multiply them with the dimensions of length ($L$), potential difference ($\Delta V$), and time ($\Delta t$).

While this brute-force method is mathematically correct, it is a classic trap designed to consume your precious time in competitive exams. But what if we could bypass this entirely?

In physics, whenever you see ${\epsilon }_0$ and a geometric length $L$ hanging out together, your mind should immediately jump to the concept of a parallel plate capacitor.

Imagine a simple capacitor with plates of area $A$ separated by a distance $L$. The geometric formula for its capacitance is:

Now, if we connect this capacitor to a battery providing a potential difference $\Delta V$, it will store a charge $\Delta q$. The electrical definition of capacitance is:

Since both expressions describe the exact same physical property of the capacitor, we can confidently equate them:

By rearranging this equation, we can isolate ${\epsilon }_0$:

Look at what we've achieved! We have expressed the complex permittivity ${\epsilon }_0$ entirely in terms of simpler, more manageable quantities.

Now, let's bring back our original intimidating expression:

We will carefully substitute our newly found expression for ${\epsilon }_0$ into this equation:

This is where the magic happens. Notice how the potential difference $\Delta V$ beautifully cancels out from the numerator and the denominator.

The expression is already looking significantly cleaner. But we are not done yet.

Let's look at the dimensions of the remaining geometric terms. What is the dimension of Area $A$? Since area is fundamentally a product of two lengths, its dimensional formula is simply $[L^2]$.

Therefore, dimensionally, the $L^2$ in the numerator and the Area $A$ in the denominator will perfectly annihilate each other!

And what exactly is the rate of flow of charge, $\frac{\Delta q}{\Delta t}$? It is the fundamental definition of electric current!

Thus, the dimensional formula for $X$ is exactly the same as that of current. A problem that initially looked like a dimensional nightmare turned out to be an elegant play of standard physics formulas. Always remember: in physics, standard formulas are your best friends for dimensional analysis!