To solve this problem, we need to calculate two things:

1. The distance between the two charges given the potential energy.
2. The initial force on the charge $-4 \times 10^{-9} \, \text{C}$.

Step 1: Distance between the charges

The potential energy $U$ between two charges is given by the formula:

$U = \frac{k \cdot q_1 \cdot q_2}{r}$

Where:

• $U = 9.5 \, \text{kJ} = 9.5 \times 10^3 \, \text{J}$ (convert to joules),
• $k = 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2$ is Coulomb's constant,
• $q_1 = -8 \times 10^{-6} \, \text{C}$,
• $q_2 = -4 \times 10^{-9} \, \text{C}$,
• $r$ is the distance between the charges.

Now, solve for $r$:

$r = \frac{k \cdot q_1 \cdot q_2}{U}$

Substituting the known values:

$r = \frac{(8.99 \times 10^9) \cdot (-8 \times 10^{-6}) \cdot (-4 \times 10^{-9})}{9.5 \times 10^3}$

Notice that both charges are negative, so their product will be positive.

Now, performing the calculation:

$r = \frac{(8.99 \times 10^9) \cdot (8 \times 10^{-6}) \cdot (4 \times 10^{-9})}{9.5 \times 10^3}$ $r = \frac{2.8768 \times 10^{-5}}{9.5 \times 10^3}$ $r = 3.03 \, \text{m}$

So, the distance between the two charges is approximately 3.03 meters.

Step 2: Force on the charge $-4 \times 10^{-9} \, \text{C}$

The force $F$ between two charges is given by Coulomb's law:

$F = \frac{k \cdot q_1 \cdot q_2}{r^2}$

We already know the values for $k$, $q_1$, $q_2$, and $r$.

Substitute the values:

$F = \frac{(8.99 \times 10^9) \cdot (8 \times 10^{-6}) \cdot (4 \times 10^{-9})}{(3.03)^2}$

Now calculate:

$F = \frac{2.8768 \times 10^{-5}}{9.1809}$ $F \approx 3.13 \times 10^{-6} \, \text{N}$

Thus, the initial force on the charge $-4 \times 10^{-9} \, \text{C}$ is approximately 3.13 µN (micro-Newtons).

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Would you like a more detailed explanation of any part of this?

Here are 5 related questions for further exploration:

1. What is the significance of Coulomb's constant $k$ in electric force and potential energy calculations?
2. How would the distance change if the potential energy between the charges were doubled?
3. How does the sign of the charges affect the force between them?
4. What is the relationship between the force and the distance between two charges?
5. How does the potential energy between charges change with distance?

Tip: Always ensure that units are consistent, especially when working with large or small charges and energies.