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A point charge is a theoretical charge located at a single point in space. This concept is useful because it simplifies the analysis of electric forces and fields. In practical terms, any particle with a negligible size compared to the distances involved in the problem can be considered as a point charge. This allows us to focus solely on the effects of its electric charge without worrying about its geometric shape or size.

In the exercise, we have two point charges, denoted as \(q_1\) and \(q_2\). \(q_1\) is stationary at the origin, while \(q_2\) moves along a path. As each charge has a significant amount of charge in microcoulombs (\( ext{μC}\)), their interaction will create noticeable electric forces that influence the movement of \(q_2\).

Using the point charge model, we can apply simplified mathematical principles to calculate the electric forces and potential energy changes as one charge influences the other.

In physics, the work done by a force is the energy transferred by that force over a distance. When it comes to electric forces, we think about how an electric field does work on a charged particle. This work changes the particle’s potential and kinetic energy.

The work done by the electric force on \(q_2\) can be found using the difference in electric potential energy as the charge moves. In this exercise, \(q_2\) moves in the electric field created by \(q_1\), and thus, it experiences a change in potential energy.

The formula used for calculating the work done, \(W\), is related to the difference in inverse distances from the point charge \(q_1\):

• \(W = k \, q_1 \, q_2 \, \left( \frac{1}{r_f} - \frac{1}{r_i} \right)\)

This equation stems from the idea that work done is the change in electric potential energy, hence correlating with the movement from one potential to another.

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electric force between two stationary point charges 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:

• \( F = k \frac{|q_1 \cdot q_2|}{r^2} \)

where:

• \(F\) represents the force between the charges
• \(k\) is 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 charges

Coulomb's Law is essential in determining how two charges interact. In this exercise, it guides the calculation of changes in potential energy due to the movement of \(q_2\) in the electric field produced by \(q_1\).

The calculation of distances in this exercise is essential for determining the change in potential energy as the point charge \(q_2\) moves. Initially, \(q_2\) is at position \((0.150, 0)\) and moves to \((0.250, 0.250)\). These distances are measured from \(q_1\) found at the origin (0,0).

Distance is calculated using the Pythagorean theorem:

• Initial distance, \(r_i = \sqrt{(x_i - 0)^2 + (y_i - 0)^2}\)
• Final distance, \(r_f = \sqrt{(x_f - 0)^2 + (y_f - 0)^2}\)

For \(q_2\)'s initial position, this results in \( r_i = 0.150 \, \text{m} \), and for the final position, \( r_f = 0.354 \, \text{m} \). These distances are critical as they are directly used in calculating the work done through the electric potential change.