To calculate the work needed to move the charges infinitely far from one another, we need to calculate the total potential energy of the system. The work required is equal to the potential energy of the configuration of the three charges.

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

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

where:

• $k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2$ is Coulomb's constant,
• $q_1$ and $q_2$ are the charges,
• $r$ is the distance between the two charges.

Step 1: Identify all pairs of charges

There are 3 charges and hence 3 unique pairs of interactions:

1. Between the top left charge ($q_1 = -6.1 \, \mu C$) and the bottom left charge ($q_2 = -3.3 \, \mu C$),
2. Between the top left charge ($q_1 = -6.1 \, \mu C$) and the top right charge ($q_3 = 2.7 \, \mu C$),
3. Between the bottom left charge ($q_2 = -3.3 \, \mu C$) and the top right charge ($q_3 = 2.7 \, \mu C$).

Step 2: Calculate distances between charges

From the diagram:

• The distance between the top left and bottom left charges is $0.16 \, \text{m}$,
• The distance between the top left and top right charges is $0.25 \, \text{m}$,
• The distance between the bottom left and top right charges can be calculated using the Pythagorean theorem: $r_{23} = \sqrt{(0.25)^2 + (0.16)^2} = \sqrt{0.0625 + 0.0256} = \sqrt{0.0881} \approx 0.297 \, \text{m}$

Step 3: Calculate potential energies for each pair

We'll now calculate the potential energy for each of the three pairs of charges.

1. Between $q_1 = -6.1 \, \mu C$ and $q_2 = -3.3 \, \mu C$: $U_{12} = \frac{k q_1 q_2}{r_{12}} = \frac{(8.99 \times 10^9) (-6.1 \times 10^{-6})(-3.3 \times 10^{-6})}{0.16}$

2. Between $q_1 = -6.1 \, \mu C$ and $q_3 = 2.7 \, \mu C$: $U_{13} = \frac{k q_1 q_3}{r_{13}} = \frac{(8.99 \times 10^9) (-6.1 \times 10^{-6})(2.7 \times 10^{-6})}{0.25}$

3. Between $q_2 = -3.3 \, \mu C$ and $q_3 = 2.7 \, \mu C$: $U_{23} = \frac{k q_2 q_3}{r_{23}} = \frac{(8.99 \times 10^9) (-3.3 \times 10^{-6})(2.7 \times 10^{-6})}{0.297}$

Step 4: Sum the potential energies

The total potential energy $U_{\text{total}}$ of the system is the sum of the potential energies for all three pairs:

$U_{\text{total}} = U_{12} + U_{13} + U_{23}$

Let me calculate these values for you.The potential energies between the pairs of charges are as follows:

1. Between $q_1 = -6.1 \, \mu C$ and $q_2 = -3.3 \, \mu C$: $U_{12} = 1.13 \, \text{J}$
2. Between $q_1 = -6.1 \, \mu C$ and $q_3 = 2.7 \, \mu C$: $U_{13} = -0.59 \, \text{J}$
3. Between $q_2 = -3.3 \, \mu C$ and $q_3 = 2.7 \, \mu C$: $U_{23} = -0.27 \, \text{J}$

The total potential energy of the system is:

$U_{\text{total}} = 0.27 \, \text{J}$

This is the work that must be done to move the three charges infinitely far from one another.

Would you like more details or have any questions about this calculation?

Here are some related questions to consider:

1. How does changing the distance between the charges affect the potential energy?
2. What would happen if all three charges were positive?
3. How is potential energy related to force between charges?
4. What role does the sign of the charge play in determining the potential energy?
5. Can you calculate the work required if one of the charges is moved only a finite distance?

Tip: Remember, when charges are of opposite sign, the potential energy is negative, indicating attraction, while for like charges, it's positive, indicating repulsion.