Two charges q1 = 7 µC, q2 = -36 µC, are L = 13 cm apart. A third charge is to be placed on the line between the two charges. How far from q1 should the third charge be placed so that the net electric force on the third charge is minimized?

To minimize the net electric force on the third charge, we need to place it at a point where the forces due to the two charges \(q_1\) and \(q_2\) cancel each other out. 

Let's denote the third charge as \(q_3\), and let \(x\) be the distance from \(q_1\) to \(q_3\). Therefore, the distance from \(q_3\) to \(q_2\) will be \(L - x\).

The magnitudes of the forces on \(q_3\) due to \(q_1\) and \(q_2\) are given by Coulomb's law:

\[
F_1 = \frac{k \cdot |q_1| \cdot |q_3|}{x^2}
\]

\[
F_2 = \frac{k \cdot |q_2| \cdot |q_3|}{(L - x)^2}
\]

where:
- \(F_1\) is the force on \(q_3\) due to \(q_1\),
- \(F_2\) is the force on \(q_3\) due to \(q_2\),
- \(k\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2\)).

For the net force to be minimized, the magnitudes of these forces must be equal:

\[
\frac{k \cdot |q_1| \cdot |q_3|}{x^2} = \frac{k \cdot |q_2| \cdot |q_3|}{(L - x)^2}
\]

We can simplify this equation by canceling out the common factors \(k\) and \(q_3\):

\[
\frac{|q_1|}{x^2} = \frac{|q_2|}{(L - x)^2}
\]

Substituting the given values:

\[
\frac{7 \, \mu\text{C}}{x^2} = \frac{36 \, \mu\text{C}}{(0.13 - x)^2}
\]

Taking the square root of both sides:

\[
\frac{\sqrt{7}}{x} = \frac{\sqrt{36}}{0.13 - x}
\]

Simplifying further:

\[
\frac{\sqrt{7}}{x} = \frac{6}{0.13 - x}
\]

Now, solve for \(x\):

\[
\sqrt{7} \cdot (0.13 - x) = 6x
\]

\[
\sqrt{7} \cdot 0.13 - \sqrt{7} \cdot x = 6x
\]

\[
\sqrt{7} \cdot 0.13 = 6x + \sqrt{7} \cdot x
\]

\[
0.13 \sqrt{7} = x (6 + \sqrt{7})
\]

\[
x = \frac{0.13 \sqrt{7}}{6 + \sqrt{7}}
\]

Now, we can calculate the numerical value of \(x\).

\[
x \approx \frac{0.13 \times 2.64575}{6 + 2.64575} \approx \frac{0.34395}{8.64575} \approx 0.0398 \text{ m} \approx 3.98 \text{ cm}
\]

So, the third charge should be placed approximately **3.98 cm** from \(q_1\) to minimize the net electric force on it.

Would you like more details or have any questions?

Here are some related questions:
1. What happens if \(q_3\) is placed exactly at the midpoint between \(q_1\) and \(q_2\)?
2. How would the position of \(q_3\) change if \(q_1\) and \(q_2\) had the same sign?
3. How does the net force on \(q_3\) vary as it moves along the line between \(q_1\) and \(q_2\)?
4. What if the third charge \(q_3\) has a specific value instead of being arbitrary?
5. How does the distance between \(q_1\) and \(q_2\) affect the position of \(q_3\) for force minimization?

**Tip:** When dealing with forces, always consider the direction and magnitude, as they can greatly influence the net result.

Learn how to minimize the net electric force on a third charge placed between two charges (7 µC and -36 µC) separated by 13 cm. This problem involves applying Coulomb's law and force vector analysis. Suitable for advanced high school students and introductory college physics.

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