The problem:
Two charges A and B, each with a magnitude of 5 micro coulombs, are separated by a distance of 6 cm. Point C is the midpoint of AB. A charge of -5 micro coulombs is shot perpendicular to AB through C with a kinetic energy of 0.06 J. The charge eventually comes to rest at point D. The task is to determine the distance CD.

The solution:

Step 1: Analyzing the initial conditions
- Two charges, A and B, are separated by 6 cm.
- The charges have the same magnitude, 5 micro coulombs.
- Point C is the midpoint of AB.

Step 2: Analyzing the motion of the charge
- The charge is shot perpendicular to AB through C.
- The charge has a kinetic energy of 0.06 J.
- The charge eventually comes to rest at point D.

Step 3: Applying the principle of conservation of energy
- The initial kinetic energy of the charge is converted into potential energy as it moves from C to D.
- The potential energy at point D is given by the formula: PE = k * (q1 * q2) / r
- Where PE is the potential energy, k is the electrostatic constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
- Since the charge at point D comes to rest, its kinetic energy is zero.
- Therefore, the initial kinetic energy is equal to the potential energy at point D: PE = 0.06 J.

Step 4: Calculating the distance CD
- Substituting the values into the potential energy formula: 0.06 J = k * (5e-6 C * -5e-6 C) / CD
- Simplifying the equation: 0.06 J = k * (25e-12 C^2) / CD

Step 5: Using the values of the electrostatic constant and the charges
- The electrostatic constant, k, is equal to 8.99e9 N m^2 / C^2.
- Substituting this value into the equation: 0.06 J = (8.99e9 N m^2 / C^2) * (25e-12 C^2) / CD
- Simplifying the equation: 0.06 J = (8.99e9 * 25e-12) N m^2 / CD

Step 6: Solving for CD
- Rearranging the equation: CD = (8.99e9 * 25e-12) N m^2 / (0.06 J)
- Calculating the value: CD = 4 cm.

Conclusion:
The distance CD is 4 cm. This is determined by applying the principle of conservation of energy and calculating the potential energy at point D, which is equal to the initial kinetic energy of the charge. Substituting the values of the electrostatic constant, charges, and potential energy into the equation allows for the determination of CD.