91Ó°ÊÓ

Coulomb's Law is a fundamental principle of electrostatics that describes the interaction between electric charges. 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. Mathematically, it is expressed as:
\[F = k \frac{|Q_1 Q_2|}{r^2}\]where:

• \(F\) is the force between the charges,
• \(k\) is Coulomb's constant (\(8.9875 \times 10^9 \text{ Nm}^2/\text{C}^2\)),
• \(Q_1\) and \(Q_2\) are the magnitudes of the charges, and
• \(r\) is the distance between the centers of the two charges.

Coulomb's Law is essential for understanding how charged objects interact. It is crucial in calculating the electrostatic forces and plays a significant role in determining equilibrium positions, such as in this exercise.

When applied to multiple charge systems, as seen in the exercise, the superposition principle is used. This means you sum the forces exerted on a charge by other charges to find the net force acting on it.

Electric charge is a fundamental property of matter, appearing in two types: positive and negative. Objects with the same type of charge repel each other, while those with opposite charges attract each other.
The unit of charge is the Coulomb (C), a measure of the amount of electricity transported by a steady current of one ampere in one second. In this problem, charges are described in nanocoulombs (nC), which is \(10^{-9}\) Coulombs.

The problem presents three charges: two negative and one positive. It's important to note that the negative charges will repel the positive one. Thus, our goal when placing the third charge is to find a position where these repulsive forces cancel out, creating a net electrostatic force of zero.

• A negative charge such as \(-3.00 \text{ nC}\) contributes to pushing the positive charge away.
• Similarly, the \(-5.80 \text{ nC}\) charge will also exert a repelling force.
• Understanding interactions between these charges helps predict the behavior and outcome of the forces at play.

In the context of this exercise, algebra is used to solve equations derived from Coulomb's Law to find equilibrium positions. We aim to determine the position \(x\), where the third charge can be placed, such that the net electrostatic force is zero.

This involves setting up expressions for the forces and solving for \(x\). When the third charge is placed between the negative charges, we equate the electrostatic forces due to each charge to each other because their net force must be zero for equilibrium.

• When calculating where \(Q_3\) should be set, let \(x\) be the distance between \(Q_3\) and \(Q_1\).
• Here, the equation \(\frac{kQ_1Q_3}{x^2} = \frac{kQ_2Q_3}{(0.5-x)^2}\) is arranged and simplified algebraically.

After formulating this equation, terms like the electrostatic constant \(k\) and the charge of \(Q_3\) are cancelled out since both sides are equal.

Next, solving involves rearranging the equation to isolate \(x\) and using techniques such as square-root extraction or quadratic formulas to find the two possible solutions for \(x\). This approach ensures that all mathematical steps are logically sound towards reaching the solution of no net force condition.