Animated Solution for Physics - Physics and Measurement: In the relation, $p=\frac{\alpha }{\beta }{e}^{-\frac{\alpha Z}{k\theta }}$ $p$ is pressure, $Z$ is distance, $k$ is Boltzmann's constant and $\theta$ is the temperature. The dimensional formula of $\beta$ will be

Imagine you are looking at a complex physical equation for the first time. It can look intimidating, right? But dimensional analysis gives us a superpower: we can break down even the most terrifying equations into simple, fundamental building blocks.

In our problem, we are given the relation $p=\frac{\alpha }{\beta }{e}^{-\frac{\alpha Z}{k\theta }}$. We need to find the dimensional formula for $\beta$.

The first thing that should catch your eye is the exponential term. A fundamental law of physics states that the argument of any exponential function must be a pure number. You cannot raise $e$ to the power of "meters" or "seconds". It must be dimensionless!

Let's use this powerful insight. We take the exponent, $\frac{\alpha Z}{k\theta }$, and set its dimension to $[M^0{L}^0{T}^0]$.

This gives us our first master equation: $\left[\frac{\alpha Z}{k\theta }\right]=[M^0{L}^0{T}^0]$.

From this, we can easily isolate the dimension of $\alpha$. By rearranging the terms, we find that $[\alpha ]=\left[\frac{k\theta }{Z}\right]$. We will keep this result safe, as it is the key to unlocking the rest of the problem.

Now, let's look back at the original equation. Since the entire exponential term $e^{-\frac{\alpha Z}{k\theta }}$ is just a number, it has no dimensions.

This means the dimension of the left side, which is pressure $p$, must be exactly equal to the dimension of the coefficient on the right side, which is $\frac{\alpha }{\beta }$.

So, we can write $[p]=\left[\frac{\alpha }{\beta }\right]$.

Our goal is to find $\beta$, so let's rearrange this to get $[\beta ]=\left[\frac{\alpha }{p}\right]$.

We are almost there! Let's substitute our earlier expression for $[\alpha ]$ into this new equation.

This gives us $[\beta ]=\left[\frac{k\theta }{Zp}\right]$. Now, we have $\beta$ entirely in terms of known physical quantities.

Let's recall the dimensions of these quantities. The term $k\theta$ represents thermal energy (since $k$ is the Boltzmann constant and $\theta$ is temperature), so its dimension is $[M{L}^2{T}^{-2}]$. The variable $Z$ is distance, so its dimension is $[L]$. Finally, $p$ is pressure (force per unit area), which has the dimension $[M{L}^{-1}{T}^{-2}]$.

Substituting these into our equation, we get:

Notice how beautifully the mass $[M]$ and time $[T^{-2}]$ terms cancel out from the numerator and denominator. The length terms also simplify, leaving us with just $[L^2]$ in the numerator.

Therefore, the final dimensional formula for $\beta$ is $[M^0{L}^2{T}^0]$.