Two point charges *q*1 and *q*2 are 3.05 m apart, and their total charge is 15 µC. Note that you may need to solve a quadratic equation to reach your answer. If one charge attracts the other with a force of 0.242 N, what are the two charges (in µC)?

To find the values of the two charges \(q_1\) and \(q_2\), we will use Coulomb's law and the fact that the total charge is given. Let's go step-by-step:

### 1. Coulomb's Law:
Coulomb’s law describes the force \(F\) between two point charges:
\[
F = k \frac{|q_1 q_2|}{r^2}
\]
where:
- \(F = 0.242 \, \text{N}\) (the force between the charges),
- \(r = 3.05 \, \text{m}\) (distance between the charges),
- \(k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\) (Coulomb's constant),
- \(q_1\) and \(q_2\) are the charges (in coulombs).

### 2. Total Charge:
We are given that the total charge is 15 µC:
\[
q_1 + q_2 = 15 \, \mu C = 15 \times 10^{-6} \, \text{C}
\]

### 3. Set Up the System of Equations:
From Coulomb's law, we can express the force equation:
\[
0.242 = \frac{(8.99 \times 10^9) |q_1 q_2|}{(3.05)^2}
\]
Also, from the total charge condition:
\[
q_1 + q_2 = 15 \times 10^{-6} \, \text{C}
\]

### 4. Solve the System:
Let’s first solve these equations step by step. I’ll compute this next.It seems that the presence of the absolute value in the Coulomb force equation complicates the direct solution. Let's approach this by solving the system without the absolute value first and assume both charges are positive. We can then evaluate the solutions considering possible signs of the charges.

I’ll rewrite the system and solve it accordingly.The solutions involve complex numbers, which is not physically meaningful for this problem. This suggests that at least one charge might be negative since the force between two charges could be attractive (opposite charges). Let’s rework the approach to consider one charge being negative, and find the appropriate real values for \( q_1 \) and \( q_2 \).The solutions still involve complex numbers, which is not physically plausible for this case. To resolve this, let's return to the assumption of opposite charges (i.e., one is positive, and the other is negative). This should provide a real-valued solution based on the given conditions.

I'll adjust the setup and recalculate.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through the steps to solve this quadratic manually. Would you like me to explain the process further?

Here are some related questions to expand on the topic:
1. How does Coulomb's law change if the charges are placed in a medium other than a vacuum?
2. What is the relationship between electric field strength and force on a charge?
3. How does the force between two charges vary if one charge is doubled?
4. How can we use vector components to analyze forces between multiple charges?
5. How would the presence of a third charge between \(q_1\) and \(q_2\) affect the force between them?

**Tip:** When solving quadratic equations, always check the discriminant to determine if the solutions are real or complex.

Two point charges q1 and q2 are 3.05 m apart, and their total charge is 15 µC. If one charge attracts the other with a force of 0.242 N, what are the two charges (in µC)?

Discover how to calculate two charges separated by 3.05 m, given a total charge of 15 µC and a force of 0.242 N using Coulomb's Law. This physics problem involves using quadratic equations and is suitable for advanced high school or college-level students.

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