91Ó°ÊÓ

Coulomb's Law is a crucial principle in understanding electric forces between charged objects. It states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is given by: \[ F = k \frac{|Qq|}{r^2} \] where:

• \(F\) is the electric force between the charges
• \(k\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\))
• \(Q\) and \(q\) are the amounts of the two charges
• \(r\) is the distance between the centers of the two charges

Coulomb's Law helps us calculate not only the force but also guides us in the calculation of electric potential energy in systems of charges.

The conservation of energy principle is fundamental in physics and tells us that energy in a closed system can neither be created nor destroyed; it can only be transformed from one form to another. In the context of electric potential energy and kinetic energy, the initial energy of a system is shared between potential energy and kinetic energy at any point in time.For point charges starting from rest, the sum of the initial kinetic energy and potential energy must equal the sum of the final kinetic and potential energies:\[ K_i + U_i = K_f + U_f \]This equation simplifies in scenarios where the initial kinetic energy is zero, which allows us to solve for the kinetic energy and thereby the velocity of a charge as it moves to different positions. It helps us understand how potential energy converts into kinetic energy as charges are moved.

Kinetic energy, the energy of motion, plays a vital role when analyzing the movement of point charges. For any object, kinetic energy can be calculated using the formula:\[ K = \frac{1}{2} mv^2 \]where:

• \(K\) is the kinetic energy
• \(m\) is the mass of the object
• \(v\) is the velocity of the object

When the second charge in our problem is released, its initial kinetic energy is zero. As it moves, part of the electric potential energy is converted into kinetic energy. By applying the conservation of energy as described above, we can rearrange the equation to solve for the velocity of the charge at different distances, revealing how fast the charge will be moving at those specified points.

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. Charge can be positive or negative, and it quantifies the ability of an object to exert electric force on another object. In the context of point charges:

• Positive charges repel each other.
• Opposite charges attract.

The amount of charge is measured in Coulombs (C). In this exercise, the charges are measured in microcoulombs (\(\mu\mathrm{C}\)), where \(1 \mu\mathrm{C} = 1 \times 10^{-6} \mathrm{C}\). Understanding the nature of electric charge is crucial for determining how charges will interact, the force they will exert on each other, and how they influence potential energies in a given system.

The interaction between point charges is the cornerstone of electrostatics. When two point charges are in proximity, they exert forces on one another. These forces arise because of their electric charges, according to Coulomb's Law. Additionally, the potential energy in a system of point charges is determined by this interaction, and can be computed using:\[ U = k \frac{Qq}{r} \]This formula shows how the potential energy changes with distance, \(r\), between charges. As the charges separate, potential energy approaches zero, which aligns with the definition of potential energy being zero at infinite separation. Analyzing point charge interactions involves understanding both the forces at play and how energy is exchanged between potential and kinetic forms as these charges move in an electric field.