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Electric charges are fundamental properties of matter that lead to electromagnetic interactions. These charges can either be positive or negative, and they dictate the nature of forces that act between charged objects.
Charged particles can exert forces over a distance, much like how magnets work, without needing to be in direct contact. Similar to how opposite magnet poles attract whereas similar poles repel, electric charges behave in the same fashion:

• Like charges (positive-positive or negative-negative) repel each other.
• Opposite charges (positive-negative) attract each other.

Understanding these basic principles is crucial when dealing with problems involving electric forces, such as the one in our original exercise. Keeping these fundamentals in mind will help make sense of how forces and charges interact.

The force between electric charges is described by Coulomb's Law, which quantifies the interaction. This law states that the force (F) between two point charges is directly proportional to the product of the magnitudes of the charges (q_1 and q_2) and inversely proportional to the square of the distance (r) between them:\[F = k \frac{q_1 q_2}{r^2}\]where k is Coulomb's constant (8.99 \times 10^9 \text{ N m}^2/\text{C}^2).

The widespread application of this formula allows you to predict how strong the repulsive or attractive force is between two charges. Importantly, the square of the distance in the formula shows us that changing the distance even slightly can have a large impact on the magnitude of the force. This inverse-square law is crucial in solving problems like the one discussed, where knowing the force helps us find unknown charges when other variables are given.

When solving problems involving equal charges, the math simplifies considerably because both charges have the same value. In these cases, each charge is simply denoted as q, and the problem becomes about finding q based on given values like distance and force.
In our exercise, when the charges are equal, the formula simplifies to:\[F = k \frac{q^2}{r^2}\]This means the force is calculated based on one variable square, reducing complexity.
For unequal charges, the scenario shifts slightly. If one charge is a multiple of the other, extra care is needed. In the problem, one charge is four times the other, thus taking on the forms q and 4q. The relationship becomes:\[F = k \frac{4q^2}{r^2}\]By understanding the relationships between charges, these complex problems become more manageable. The operations involve more algebra, but the principles and steps remain grounded in fundamental physics.

To solve any physics problem related to Coulomb's Law effectively, a structured approach is essential. Let’s outline the steps based on the given problem:

• Identify knowns and unknowns: Begin by gathering what is already given in the problem statement: the distance ( r ) between the charges and the force ( F ) are known here.
• Set up Coulomb’s Law: Use the law as your equation base. Choose appropriate formulas based on whether the charges are equal or unequal.
• Solve for the unknown: Rearrange the formula to solve for the unknown variable. With equal charges, rearrange to find q . With unequal charges, you must consider the relationship between the charges, like q and 4q .
• Substitute and calculate: Insert all known values into the rearranged formula and carefully compute to find your answer.

Following these steps not only provides discipline but also reduces errors, ensuring you've correctly applied physical laws to reach a conclusion. Efficient problem-solving in physics builds confidence and deepens comprehension of fundamental concepts like electrodynamics.