Animated Solution for Physics - Physics and Measurement: In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {ϵ}_0}$, where ${ϵ}_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[e{]}^{\alpha }[m_e{]}^{\beta }[h{]}^{\gamma }[k{]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ is ________.

Beyond Just Checking Units

Imagine a universe where we don't measure things in meters, kilograms, or seconds. Instead, we use the fundamental constants of nature—like the speed of light, Planck's constant, or the charge of an electron—as our basic yardsticks. This isn't just science fiction; physicists use these "natural units" all the time to simplify complex equations in quantum mechanics, particle physics, and general relativity. By setting constants like $c$ or $h$ to 1, equations become incredibly elegant, revealing the pure geometric and physical relationships without the clutter of human-invented units.

In this fascinating JEE Advanced problem, we are asked to step into one such custom system of units. We need to express the magnetic field $B$ in terms of the electron charge $e$, electron mass $m_e$, Planck's constant $h$, and Coulomb's constant $k$. This is a classic test of your mastery over Dimensional Analysis and the Principle of Dimensional Homogeneity. It challenges you not just to remember formulas, but to manipulate the very fabric of physical dimensions.

The Building Blocks of Physics

Before we can build the master equation, we need to know our building blocks. We must find the standard SI dimensional formula for every quantity involved. Let's break them down one by one, understanding the physics behind each.

First, let's look at the target quantity: the magnetic field $B$. If you ever forget its dimension, don't panic. Just recall the Lorentz force equation, which describes the force on a moving charge in a magnetic field:

Rearranging this equation to solve for $B$, we get $B=\frac{F}{qv}$. Now, we substitute the fundamental dimensions. Force is mass times acceleration ($ML{T}^{-2}$), charge is current times time ($AT$), and velocity is length over time ($L{T}^{-1}$). Plugging these in, we find:

Next, let's analyze our new "base" quantities. These are the fundamental constants that will form our new coordinate system of units:

1. Electric Charge ($e$): In the SI system, electric current ($A$, Ampere) is considered a fundamental base quantity, not charge. Therefore, charge is dimensionally represented as current multiplied by time.

2. Electron Mass ($m_e$): This is the simplest one. Mass is a fundamental dimension in mechanics.

3. Planck's Constant ($h$): This constant is the bridge between the macroscopic and quantum worlds. Recall the energy of a photon given by the Planck-Einstein relation, $E = h u$. Since frequency $ u$ has the dimension of inverse time ($T^{-1}$), $h$ must have the dimension of Energy divided by frequency. Energy is work done, which is force times distance ($M{L}^2{T}^{-2}$).

4. Coulomb's Constant ($k$): This constant dictates the strength of the electrostatic force between two charges. From Coulomb's Law, $F=k\frac{q_1{q}_2}{r^2}$, we can isolate $k$: Substituting the dimensions of force, distance squared, and charge squared, we get:

The problem states that the magnetic field can be expressed as a product of these constants raised to unknown powers:

This is where the Principle of Dimensional Homogeneity comes into play. It is a non-negotiable law of physics stating that for any valid physical equation, the dimensions on the left-hand side must perfectly match the dimensions on the right-hand side. You cannot equate apples to oranges, and you cannot equate a force to an energy. Let's substitute our derived dimensions into this framework:

Now, we must carefully expand and group the terms on the right side by their base quantities ($M,L,T,A$). This requires meticulous algebra. A single dropped negative sign here will ruin the entire calculation.

A Strategic Approach

By equating the exponents of corresponding base quantities on both sides, we generate a system of four linear equations. This is the mathematical heart of the problem:

For $M$: $\beta +\gamma +\delta =1$ For $L$: $2\gamma +3\delta =0$ For $T$: $\alpha -\gamma -4\delta =-2$ For $A$: $\alpha -2\delta =-1$

At first glance, a system of four equations with four variables might look intimidating. You might be tempted to set up a massive matrix and use Cramer's rule. But in competitive exams like JEE, there is almost always a faster, more elegant path. The trick is to look for the simplest relationships—equations with the fewest variables.

Notice the equation for $L$. It only contains $\gamma$ and $\delta$:

We also have a very simple relation from the $A$ equation, linking $\alpha$ directly to $\delta$:

Now, we have both $\alpha$ and $\gamma$ expressed entirely in terms of $\delta$. We can substitute these expressions into the dense $T$ equation to solve for $\delta$:

Let's carefully expand this:

Combine the $\delta$ terms:

Add 1 to both sides:

With $\delta$ unlocked, the rest of the dominoes fall instantly. We just plug $\delta =2$ back into our earlier simple relations:

Finally, we substitute the known values of $\gamma$ and $\delta$ into the $M$ equation to find our last variable, $\beta$:

We have successfully navigated the algebraic maze and found all the exponents: $\alpha =3$, $\beta =2$, $\gamma =-3$, and $\delta =2$.

The question asks for the simple arithmetic sum of these values:

This problem beautifully illustrates how dimensional analysis is not just a mundane tool for checking if an equation is right or wrong. It is a powerful mathematical framework that can reveal deep, hidden relationships between fundamental constants. It shows us that the magnetic field is intimately connected to the charge, mass, quantum nature (Planck's constant), and electrostatic properties of the universe.

Always ensure you are rock-solid on deriving dimensions from basic formulas. Do not memorize complex dimensional formulas; instead, memorize the fundamental physics equations (like Lorentz force or Coulomb's law) and derive the dimensions on the fly. This approach is foolproof and will serve you well in the high-pressure environment of the JEE Advanced!