91Ó°ÊÓ

Coulomb's Law is fundamental in understanding electrostatics. It describes the force between two charged objects. The law states that the electrostatic force (\( F \)) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where:

• \( k \) is the Coulomb's constant \( (8.99 \times 10^9 \text{ Nm}^2/\text{C}^2) \).
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges.
• \( r \) is the distance between the centers of the two charges.

For protons, both charges are equal (\( 1.602 \times 10^{-19} \text{ C} \)), making these forces repulsive. This foundational law helps calculate how charges interact and is a building block for deeper electrostatic concepts.

The concept of work done in electric fields involves moving charges under the influence of an electrostatic force. Similar to mechanical work, it refers to the energy needed to move a charge against the electric force. Here, work done (\( W \)) by an electric field in moving a charge from one point to another is given by:
\[ W = \int_{r_1}^{r_2} F \, dr = \int_{r_1}^{r_2} \frac{k \cdot e^2}{r^2} \, dr \]When you solve this integral, you find:
\[ W = k \cdot e^2 \left( \frac{1}{r_2} - \frac{1}{r_1} \right) \]This expression calculates the work required to bring two protons closer or push them apart. In our problem, we are moving protons from a distance of \( r_1 = 2.00 \times 10^{-10} \text{ m} \) to \( r_2 = 3.00 \times 10^{-15} \text{ m} \). The work done is the energy needed to overcome the electrostatic repulsion between them.

The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the context of the proton exercise, it helps us understand how potential energy converts into kinetic energy.
The initial potential energy when protons are at their closest (\( r_2 \)) equals the work done to bring them from \( r_1 \) to \( r_2 \). When released, this potential energy becomes kinetic energy, causing the protons to accelerate away from each other. The equation for kinetic energy conversion is:
\[ 2 \cdot \frac{1}{2} mv^2 = W \]Simplifying gives:
\[ mv^2 = W \]Here, \( m \) is the mass of a proton (\( 1.67 \times 10^{-27} \text{ kg} \)). With known work done (\( 7.68 \times 10^{-14} \text{ J} \)), you can find the speed \( v \) using the rearranged formula:
\[ v = \sqrt{\frac{W}{m}} \]This formula allows us to calculate the speed of protons when they are released from rest.

Protons are positively charged particles found in atomic nuclei, and they interact with each other through electrostatic forces. This interaction is crucial in understanding nuclear forces and structures. In our exercise, two protons start at a typical atomic distance of \( 2.00 \times 10^{-10} \text{ m} \) and are brought closer to a typical nuclear distance of \( 3.00 \times 10^{-15} \text{ m} \).
As they are both positively charged, they repel each other. This repulsion must be overcome to bring them closer, requiring work to be done against this electrostatic force. The work done in this process is part of the energy concepts in electrostatics, illustrating how energy considerations are crucial for understanding nuclear interactions. Understanding this interaction helps explain not only atomic structures but also phenomena like nuclear fusion, where such close proton-proton interactions occur.