The problem asks you to find the point between two charges $q_1$ and $q_2$ where the total electric potential is zero. The charges and positions are given as:

• $q_1 = 2.5 \, \text{C}$ at $x = 0 \, \text{m}$
• $q_2 = -8.9 \, \text{C}$ at $x = 12 \, \text{m}$

The electric potential due to a point charge $q$ at a distance $r$ is given by:

$V = \frac{kq}{r}$

where $k$ is Coulomb's constant, and $r$ is the distance from the charge to the point of interest.

To find the location where the total potential is zero, we set the potentials from the two charges equal in magnitude but opposite in sign:

$\frac{kq_1}{x} = \frac{kq_2}{12 - x}$

Simplifying (since $k$ cancels out):

$\frac{q_1}{x} = \frac{|q_2|}{12 - x}$

Substitute the values of $q_1$ and $q_2$:

$\frac{2.5}{x} = \frac{8.9}{12 - x}$

Now we solve for $x$.

Let me calculate this.The point where the total potential is zero is at $x \approx 2.63 \, \text{m}$ from charge $q_1$. This is the location along the number line between the two charges.

Would you like more details on this solution, or do you have any further questions?

Here are five related questions you might consider:

1. How does the distance between the charges affect the position where the potential is zero?
2. What happens to the point where the potential is zero if both charges have the same sign?
3. Can you calculate the electric field at the point where the potential is zero?
4. How would the problem change if one charge were moved closer to the other?
5. How does the magnitude of each charge influence the potential at different points?

Tip: When solving for electric potential, remember that potentials add algebraically, unlike electric fields that are vectors.