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In electrostatics, potential energy refers to the energy a charged object possesses due to its position relative to other charged objects. When dealing with point charges, potential energy is the result of the interplay between electrostatic forces.
For two point charges, the potential energy (U) can be calculated using the formula:\[ U = \frac{1}{4\pi \varepsilon_0} \cdot \frac{q_1 q_2}{r} \]where:

• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance separating them,
• \( \varepsilon_0 \) is the permittivity of free space.

The sign of the potential energy tells us about the nature of the force. If negative, like in this example, it indicates an attractive force between charges of opposite signs. The potential energy depends on how far apart the charges are and their magnitudes.

Escape velocity is the minimum speed needed for an object to "break free" from the gravitational or electrostatic attraction of another object, without any external force contributing after the initial thrust.
In the context of electrostatics, it refers to the speed necessary for a charged particle to leave the electric field of another without being pulled back.
In the given problem, a sphere with mass needs to overcome the attractive force exerted by another charged sphere. Its potential energy must be transformed entirely into kinetic energy at the point of release, while ensuring that the velocity becomes zero at infinite distance:
\[ \frac{1}{2} mv^2 = |U| \]
Here, \( m \) is the mass of the sphere, \( v \) is the escape velocity, and \( U \) is the potential energy. By equating, you can solve for the velocity the sphere requires at initiation.

Point charges are idealized charges situated at a single point in space. This simplification allows physicists and students to easily calculate important interactions without worrying about the complexities introduced by larger objects.
In practice, real objects can often approximate point charges quite well, if the distances involved are much larger than the size of the objects.
The concept is crucial in understanding and calculating forces and potential energy when analyzing electrostatic scenarios. Point charges help to explain the way charged objects interact through their respective electric fields, as well as how these interactions can affect motion and energy exchange, as in this exercise.

The electric force is a fundamental force that occurs between charged objects. It can be attractive or repulsive, depending on the nature of the charges involved.
Opposite charges, such as negative and positive, attract each other. Like charges repel.
The electric force between two point charges can be described by Coulomb's Law:\[ F = \frac{1}{4\pi \varepsilon_0} \cdot \frac{q_1 q_2}{r^2} \]where:

• \( q_1 \) and \( q_2 \) are the charges,
• \( r \) is the distance between them,
• \( \varepsilon_0 \) is the permittivity of free space.

This law highlights how the force weakens as charges move farther apart, an inverse square relationship. Understanding electric force is key for exploring how charged objects behave and interact over distances in electrostatic fields.