Animated Solution for Physics - Electromagnetic Waves: A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E}\times \vec{B})/{\mu }_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and ${\mu }_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity (ies) ?

Imagine you are standing outside on a bright, sunny day. You feel the warmth of the sun on your skin. What you are experiencing is the transfer of energy from the sun to the Earth through the vacuum of space.

This energy travels in the form of electromagnetic waves. But how do we measure the rate at which this energy is flowing? Enter the Poynting vector, denoted by $\vec{S}$.

In this problem, we are given the mathematical definition of the Poynting vector: $\vec{S}=\frac{\vec{E}\times \vec{B}}{{\mu }_0}$. Here, $\vec{E}$ is the electric field, $\vec{B}$ is the magnetic field, and ${\mu }_0$ is the permeability of free space. The question asks us to find the dimensions of $\vec{S}$ and match them with the given options.

Before we dive into a sea of dimensional formulas, let's take a step back and understand what $\vec{S}$ actually represents physically. The Poynting vector is not just a random combination of fields; it has a profound physical significance.

It represents the directional energy flux of an electromagnetic field. In simpler terms, it tells us the amount of energy passing through a unit area per unit time. Mathematically, we can write this as:

Now, we know that the rate of transfer of energy (Energy / Time) is defined as Power. Therefore, the expression simplifies beautifully to:

This is a massive shortcut! Instead of finding the individual dimensions of $\vec{E}$, $\vec{B}$, and ${\mu }_0$ and plugging them into the cross-product formula, we can simply find the dimensions of Power per unit Area.

Let's calculate the dimensions of $\vec{S}$ using our newfound shortcut. We know the dimensional formula for Power. Power is work done per unit time, and work is force times displacement.

The dimensional formula for Area is simply length squared:

Now, we divide the two to find the dimensions of the Poynting vector:

So, the dimensional formula for $\vec{S}$ is ${\text{M T}}^{-3}$. Notice how the length dimension completely vanishes! This is a crucial piece of information that we will use to evaluate our options.

Now comes the fun part. We need to act like detectives and investigate each option to see which one matches our target dimension of ${\text{M T}}^{-3}$. Since this is a multiple-correct question (a favorite format in JEE Advanced), we must patiently check every single option. Don't rush through this!

The first option is $\frac{\text{Energy}}{\text{charge}\times \text{current}}$. Let's break it down.

Plugging these in:

This clearly does not match ${\text{M T}}^{-3}$. So, Option (A) is incorrect.

The second option is $\frac{\text{Force}}{\text{Length}\times \text{Time}}$. Let's see what this yields.

Substituting the values:

Oh wow! The $\text{L}$ cancels out perfectly, leaving us with exactly ${\text{M T}}^{-3}$. This matches our target dimension perfectly. So, Option (B) is correct.

The third option is $\frac{\text{Energy}}{\text{Volume}}$. This is a very famous quantity known as Energy Density.

Dividing them gives:

This is not ${\text{M T}}^{-3}$. So, Option (C) is incorrect.

The final option is $\frac{\text{Power}}{\text{Area}}$. Wait a minute, does this look familiar?

Yes! This is exactly the physical definition of the Poynting vector that we established at the very beginning.

Dividing them gives:

This is a perfect match. So, Option (D) is also correct.

What if you were sitting in the exam hall and completely blanked out on the physical meaning of the Poynting vector? Don't panic. You could still solve this problem using the "brute force" method. It involves a bit more heavy lifting, but it gets you to the exact same destination.

You would need to recall or derive the dimensions of the electric field $\vec{E}$, the magnetic field $\vec{B}$, and the permeability ${\mu }_0$.

Now, substitute these into the given formula $\vec{S}=\frac{\vec{E}\times \vec{B}}{{\mu }_0}$:

Let's carefully combine the terms in the numerator:

Now, cancel out the common terms. The ${\text{A}}^{-2}$ cancels out. One $\text{M}$ cancels out. The $\text{L}$ cancels out. And ${\text{T}}^{-5}$ divided by ${\text{T}}^{-2}$ leaves ${\text{T}}^{-3}$.

As you can see, the brute force method confirms our earlier result. However, recognizing the physical meaning of the formula saved us a significant amount of time and reduced the chances of making a silly algebraic mistake.

In competitive exams like JEE Advanced, time is your most precious resource. Always look for the elegant, conceptual path before resorting to brute force calculation.

This problem is a beautiful reminder that physics is not just about memorizing formulas; it's about understanding what those formulas represent in the real world. By recognizing that the Poynting vector represents energy flux (Power per unit Area), we turned a potentially tedious dimensional analysis problem into a quick and elegant conceptual exercise. Keep building that physical intuition!