Question

Two protons are separated by 1 x 10 m. How does the electric force compare to the gravitational force between them?

Answer 1

The electric force between two protons is significantly stronger than the gravitational force between them.

Step-by-step explanation:

In order to compare the electric force to the gravitational force between two protons, we can use Coulomb's Law for the electric force and Newton's Law of Universal Gravitation for the gravitational force.

Coulomb's Law states that the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

The formula for Coulomb's Law is given by:

`\[ F_{\text{electric}} = (k \cdot q_1 \cdot q_2)/(r^2) \]`

where \( F_{\text{electric}} \) is the electric force,
`\( k \)`is Coulomb's constant
`(\( 8.99 * 10^9 \, \text{N m}^2/\text{C}^2 \)), \( q_1 \) and \( q_2 \)` are the charges of the particles, and \( r \) is the separation distance.

On the other hand, Newton's Law of Universal Gravitation is given by:

`\[ F_{\text{gravitational}} = (G \cdot m_1 \cdot m_2)/(r^2) \]`

where \( F_{\text{gravitational}} \) is the gravitational force,
`\( G \)` is the gravitational constant
`(\( 6.67 * 10^(-11) \, \text{N m}^2/\text{kg}^2 \))`,
`\( m_1 \) and \( m_2 \)`are the masses of the particles, and
`\( r \)`is the separation distance.

In the case of two protons, the charges are the same
`(\( q_1 = q_2 = 1.602 * 10^(-19) \, \text{C} \))` and the masses are approximately the same
`(\( m_1 = m_2 \approx 1.6726 * 10^(-27) \, \text{kg} \)).`

Given a separation distance
`\( r = 1 * 10^(-10) \, \text{m} \)`, we can compare the two forces.

Calculating both forces, we find that the electric force is approximately
`( 2.3 * 10^(36) \)`) times stronger than the gravitational force.

This vast difference in magnitude underscores the dominance of the electric force at the atomic and subatomic levels.