**Hint:** To check if a ratio is dimensionless, you need to analyze the dimensions of the quantities involved. In this case, you will use Coulomb's constant ($k_e$), Newton's gravitational constant ($G$), the mass of an electron ($m_e$), and the mass of a proton ($m_p$). You can find these constants in a Table of Physical Constants.

**Step-by-Step Solution:**

**Step 1:** Determine the Dimensionality of the Ratio:

Coulomb's constant ($k_e$), has dimensions of $\dfrac{\text{N m}^2}{\text{C}^2}$.

Newton's gravitational constant ($G$) has dimensions of $\dfrac{\text{m}^3}{\text{kg s}^2}$.

The mass of an electron ($m_e$) has dimensions of $\text{kg}$.

The mass of a proton ($m_p$) also has dimensions of $\text{kg}$.

Now, let's analyze the dimensions of the ratio $\dfrac{k_e^2}{G \cdot m_e \cdot m_p}$.

**Step 2:** Calculate the Value of the Ratio:

Using the values of the constants from a Table of Physical Constants:

$k_e \approx 8.988 \times 10^9 \, \dfrac{\text{N m}^2}{\text{C}^2}$

$G \approx 6.674 \times 10^{-11} \, \dfrac{\text{m}^3}{\text{kg s}^2}$

$m_e \approx 9.109 \times 10^{-31} \, \text{kg}$

$m_p \approx 1.673 \times 10^{-27} \, \text{kg}$

Now, calculate the value of the ratio $\dfrac{k_e^2}{G \cdot m_e \cdot m_p}$.

**Step 3:** Significance of the Ratio:

The ratio $\dfrac{k_e^2}{G \cdot m_e \cdot m_p}$ quantifies the relative strength of electromagnetic interactions (governed by Coulomb's law) to gravitational interactions (governed by Newton's law of gravitation) between an electron and a proton.

**Note:**

The dimensionality analysis confirms whether a ratio is dimensionless or not.

The numerical value of the ratio can be calculated based on the values of the constants from a Table of Physical Constants.

The ratio signifies the vast difference in the strength of electromagnetic and gravitational forces, which is a fundamental aspect of the behavior of matter at atomic and subatomic scales.