Problem:
We have three point charges Q, 2Q, and -3Q placed at the vertices of an equilateral triangle ABC. We need to find the amount of work done in shifting these charges to the midpoints A', B', and C' of the sides of the triangle.

Solution:

Step 1: Analyzing the initial configuration
In the initial configuration, the charges Q, 2Q, and -3Q are placed at the vertices of an equilateral triangle ABC. Let's analyze the forces between these charges.

- The force between Q and 2Q can be calculated using Coulomb's law:
F(1-2) = k * (Q * 2Q) / d^2
= 2kQ^2 / d^2

- Similarly, the force between Q and -3Q is:
F(1-3) = k * (Q * -3Q) / d^2
= -3kQ^2 / d^2

- The force between 2Q and -3Q is:
F(2-3) = k * (2Q * -3Q) / d^2
= -6kQ^2 / d^2

- As the triangle ABC is equilateral, the distances between the charges are equal. Therefore, d is the same for all forces.

Step 2: Analyzing the final configuration
In the final configuration, the charges are shifted to the midpoints A', B', and C' of the sides of the triangle. Let's analyze the forces between these charges in the final configuration.

- The force between Q and 2Q in the final configuration is:
F'(1-2) = k * (Q * 2Q) / (d/2)^2
= 8kQ^2 / d^2

- Similarly, the force between Q and -3Q in the final configuration is:
F'(1-3) = k * (Q * -3Q) / (d/2)^2
= -18kQ^2 / d^2

- The force between 2Q and -3Q in the final configuration is:
F'(2-3) = k * (2Q * -3Q) / (d/2)^2
= -36kQ^2 / d^2

Step 3: Calculating the work done
The work done in shifting the charges from the initial to the final configuration can be calculated using the formula:

Work = ∑ (F' * d')

- For the force between Q and 2Q, the displacement is d/2. So the work done is:
W(1-2) = F'(1-2) * (d/2)
= (8kQ^2 / d^2) * (d/2)
= 4kQ^2 / d

- Similarly, the work done for the other two forces can be calculated as:
W(1-3) = -9kQ^2 / d
W(2-3) = -18kQ^2 / d

- Adding up the work done for all forces, we get the total work