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Understanding the electric force calculation involves applying Coulomb's Law, a fundamental principle in physics that quantifies the amount of force between two charged objects. In essence, like charges repel each other, while opposite charges attract, with the force magnitude directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The formula to express this relationship is \( F_e = \frac{k_e \cdot |q1 \cdot q2|}{r^2} \), where \( F_e \) represents the electric force, \( k_e \), known as Coulomb's constant, is \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \), \( q1 \) and \( q2 \) are the electrical charges of the objects (in this case, protons), and \( r \) is the distance separating them. Protons have a charge of \( +1e \) each, where \( e \) is the elementary charge and equals \( 1.6 \times 10^{-19} \text{C} \). By substituting these known values into the formula, one can calculate the electric force exerted between two protons.

The concept of gravitational force calculation can be illustrated by Newton's universal law of gravitation. This law states that any two objects with mass will attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. This is similar to the concept of electric forces but involves mass instead of charge.

The mathematical formula for gravitational force is \( F_g = \frac{G \cdot m1 \cdot m2}{r^2} \), where \( F_g \) is the gravitational force, \( G \) is the gravitational constant which is \( 6.674 \times 10^{-11} \text{ N} \left(\text{m}^2/\text{kg}^2\right) \) , \( m1 \) and \( m2 \) are the masses of the two objects, and \( r \) remains the distance between the centers of mass of the two objects. By replacing these values with those corresponding to the masses of the protons and the distance between them, you arrive at the gravitational force between them, which is significantly weaker than the electric force.

Forces in physics are the agents of change; they can cause objects to accelerate, decelerate, remain in place, or change their state of motion in any way. There are a variety of forces that act in the universe, with gravity and electromagnetism being two of the fundamental interactions. In the context of our problem involving protons, these two forces demonstrate two key principles of physics: electromagnetism through Coulomb's Law and gravity through Newton's Law of Gravitation.

Electric forces result from properties of particles called charges, while gravitational forces are a consequence of their mass. Their respective formulas show how these forces diminish with increasing distance, known as an inverse-square law relationship. Although these forces are omnipresent, their relative strengths can differ dramatically, with electric forces being significantly stronger than gravitational forces at the subatomic scale. This relative strength is critical when analyzing the effects of these forces on matter across different scales and scenarios.