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.