Compare the electric and gravitational forces between two electrons or between two protons a certain distance apart. Why do gravitational forces predominate at astronomical distances and electric forces at atomic distances?

Gravitational force is a fundamental interaction that attracts two objects with mass. To understand how it's quantified, we use the gravitational force equation:

\[\begin{equation} F_G = G \frac{m_1m_2}{r^2} \end{equation}\]
Here, \( F_G \) represents the gravitational force between two masses, \( G \) is the gravitational constant (6.674 \times 10^{-11} N (m/kg)^2 ), \( m_1 \) and \( m_2 \) are the individual masses of the objects, and \( r \) stands for the distance between their centers. In essence, this formula tells us that the gravitational force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them. Important to note, gravity is always attractive, drawing masses toward each other. This universal force is weak compared to electric forces, but it becomes dominant at large (astronomical) distances because it's not canceled out, unlike electric forces that can be due to the presence of equal amounts of positive and negative charges.

In contrast to gravity, the electric force can both attract and repel, dictated by the presence of electrical charges. This force is tremendously stronger at close ranges than gravitational forces. It is encapsulated by Coulomb's Law, given as:

\[\begin{equation} F_E = k \frac{q_1q_2}{r^2} \end{equation}\]
In the equation, \( F_E \) represents the electric force between two charges, \( k \) signifies the electrostatic constant (8.987 \times 10^9 N m^2/C^2 ), \( q_1 \) and \( q_2 \) indicate the amounts of the charges, and \( r \) is the distance between the charges. The sign of \( q_1 \) and \( q_2 \) determines the nature of the force: like charges repel, opposite charges attract. At microscopic (atomic) distances, this force is highly significant because atoms interact through their electrons and protons, which carry charge. However, at large, astronomical scales, where bodies are generally electrically neutral, the electric forces cancel out, leaving gravity as the primary force.

Protons are subatomic particles that reside in the nucleus of an atom, characterized by their mass and charge. The mass of a proton is very small but is integral in determining the gravitational force. It has a value of:

\[\begin{equation} m_p = 1.67 \times 10^{-27} kg \end{equation}\]
Moreover, protons carry a positive electric charge, and in terms of magnitude, this charge is equal to that of an electron but opposite in polarity. The charge of a proton is:

\[\begin{equation} q_p = 1.6 \times 10^{-19} C \end{equation}\]
This charge is considered the fundamental unit of electric charge, which means it's also a building block for how we measure electric force. Since every electron has an equal but negative charge, their interactions with protons result in the strong forces that hold atoms together. Both mass and charge play pivotal roles in the exercises to compare gravitational and electric forces, enabling us to understand why different forces prevail at varying scales in the universe.