Dimensional Analysis of Fundamental Constants

Problem: Finding the Combination with Length Dimension

This question tests our understanding of dimensional analysis, a technique used in physics to check the consistency of equations and derive relationships between physical quantities. We are asked to identify which combination of the Planck constant ($h$), the speed of light ($c$), and the gravitational constant ($G$) has the fundamental dimension of length, denoted as $[L^1]$.

Understanding Physical Dimensions

Every physical quantity can be expressed in terms of fundamental dimensions like Mass ($M$), Length ($L$), and Time ($T$). Understanding these dimensions is crucial for solving problems involving physical constants.

Given Constants and Their Dimensions

The dimensions of the fundamental constants provided are:

• Planck constant ($h$): $[ML^2T^{-1}]$
• Speed of light ($c$): $[LT^{-1}]$
• Gravitational constant ($G$): $[M^{-1}L^3T^{-2}]$

Our goal is to find a combination of these constants whose resulting dimension is simply $[L]$.

Step-by-Step Dimensional Calculation for Each Option

We will now analyze each option by multiplying or dividing the dimensions of the given constants according to the expression in the option.

Analysis of Option 1: $h c G$

Let's find the dimension of the product $h \times c \times G$. We substitute the dimensions:

Combine the powers for each dimension ($M$, $L$, $T$):

This dimension is not $[L^1]$.

Analysis of Option 2: $\frac{hc^2}{G}$

First, let's find the dimension of $c^2$:

Now, let's find the dimension of the expression $\frac{hc^2}{G}$:

Calculate the dimension of the numerator $hc^2$:

Now divide by the dimension of $G$:

This dimension is not $[L^1]$.

Analysis of Option 3: $\frac{c^3}{hG}$

First, find the dimension of $c^3$:

Next, find the dimension of the product $hG$:

Now, calculate the dimension of the expression $\frac{c^3}{hG}$:

This dimension is not $[L^1]$.

Analysis of Option 4: $\sqrt{\frac{hG}{c^3}}$

We need to find the dimension of the expression inside the square root first, which is $\frac{hG}{c^3}$.

From the analysis of Option 3, we already found:

• $[hG] = [L^5T^{-3}]$
• $[c^3] = [L^3T^{-3}]$

Now, let's find the dimension of the fraction $\frac{hG}{c^3}$:

Finally, we take the square root of this dimension:

This matches the required dimension of length.

Conclusion on Length Dimension

By performing dimensional analysis on all the given options, we found that only the combination $\sqrt{\frac{hG}{c^3}}$ results in the dimension of length, $[L^1]$. This combination is known as the Planck length, a fundamental scale in quantum gravity.