Animated Solution for Physics - Physics and Measurement: In terms of potential difference $V$, electric current $I$, permittivity ${\epsilon }_0$, permeability ${\mu }_0$ and speed of light $c$, the dimensionally correct equations is/are

Dimensional analysis is one of the most powerful tools in a physicist's arsenal. It acts as a universal lie detector for equations. If the dimensions on the left side don't match the right side, the equation is undeniably wrong.

In this classic JEE Advanced problem, we are asked to verify the dimensional correctness of four equations involving potential difference ($V$), electric current ($I$), permittivity (${\epsilon }_0$), permeability (${\mu }_0$), and the speed of light ($c$).

The most obvious way to solve this is to write down the dimensional formula for every single quantity: - $[V]=[{\text{M L}}^2{\text{T}}^{-3}{\text{A}}^{-1}]$ - $[I]=[\text{A}]$ - $[{\epsilon }_0]=[{\text{M}}^{-1}{\text{L}}^{-3}{\text{T}}^4{\text{A}}^2]$ - $[{\mu }_0]=[{\text{M L T}}^{-2}{\text{A}}^{-2}]$ - $[c]=[{\text{L T}}^{-1}]$

You could then substitute these massive blocks of letters into each option and check if the left-hand side equals the right-hand side. While this method works, it is tedious, time-consuming, and highly prone to silly calculation errors. In a high-pressure exam like JEE, we need a smarter approach.

Instead of treating this as a pure math problem, let's think like a physicist. Where have we seen terms like ${\mu }_0{I}^2$ and ${\epsilon }_0{V}^2$ before?

They appear in the formulas for the energy stored in an inductor and a capacitor!

To make these useful, we need to express the inductance ($L$) and capacitance ($C$) in terms of ${\mu }_0$ and ${\epsilon }_0$. Let's use the formulas for a long solenoid and a parallel plate capacitor:

Notice a beautiful symmetry here? Both formulas contain an Area term ($A$) in the numerator and a length term ($l$ or $d$) in the denominator.

The Energy Connection

Let's substitute our expressions for $L$ and $C$ into the energy equations.

For the inductor: Rearranging this to isolate ${\mu }_0{I}^2$, we get:

We know that Work (Energy) is Force $\times$ Distance. Therefore, Energy divided by Length gives the dimension of Force.

Now, let's do the same for the capacitor: Rearranging to isolate ${\epsilon }_0{V}^2$:

This is also Force! Since both ${\mu }_0{I}^2$ and ${\epsilon }_0{V}^2$ have the exact same dimensions (Force), option (a) is dimensionally correct.

The Flow of Charge

Now let's look at option (c), which relates current ($I$), voltage ($V$), and ${\epsilon }_0$. We know that charge and voltage are related by capacitance:

And current is simply the rate of flow of charge:

Substitute our geometric formula for capacitance ($C=\frac{{\epsilon }_{0}A}{d}$):

Now, let's look closely at the geometric term $\frac{A}{d\cdot t}$. What are its dimensions? - Area ($A$) has dimension $[{\text{L}}^2]$ - Distance ($d$) has dimension $[\text{L}]$ - Time ($t$) has dimension $[\text{T}]$

This is exactly the dimension of velocity! And what is $c$? The speed of light, which is also a velocity. Therefore, dimensionally, we can replace $\frac{A}{d\cdot t}$ with $c$:

This perfectly matches option (c)!

By using physical intuition and known formulas, we elegantly proved that options (a) and (c) are dimensionally correct without ever writing down a single $[{\text{M}}^{-1}{\text{L}}^{-3}{\text{T}}^4{\text{A}}^2]$. Options (b) and (d) can be quickly dismissed as they contradict these established relationships.

Always remember: math is the language of physics, but physical intuition is the guide!