Two insulated charged copper spheres of identical size have charge qA and qB respectively. A third sphere C of the same size but uncharged is brought in contact with first and then in contact with the second and finally removed from both. What are the new charge on A and B?

To understand how the charges on the copper spheres change after bringing an uncharged sphere into contact with them, we need to consider the principles of charge distribution and conservation. When two conductive objects come into contact, charge will redistribute between them until they reach the same electric potential. Let's break this down step by step.

Initial Charges on the Spheres

We start with two insulated copper spheres, A and B, which have initial charges of qA and qB, respectively. Sphere C is uncharged at the beginning, meaning it has a charge of 0.

Step 1: Contact with Sphere A

When sphere C comes into contact with sphere A, the charges will redistribute between them. Since both spheres are identical in size, they will share the total charge equally. The total charge when they touch is:

• Total charge = qA + 0 = qA

After they touch, the charge on each sphere will be:

• Charge on A = Charge on C = (qA + 0) / 2 = qA / 2
• Charge on C = qA / 2

Step 2: Contact with Sphere B

Next, sphere C, now with a charge of qA / 2, is brought into contact with sphere B. Again, the charges will redistribute. The total charge when they touch is:

• Total charge = qB + (qA / 2)

After they touch, the charge on each sphere will be:

• Charge on B = Charge on C = (qB + qA / 2) / 2
• Charge on C = (qB + qA / 2) / 2

Final Charges on Spheres A and B

Now, let's summarize the final charges on spheres A and B after both interactions:

• Final charge on sphere A = qA / 2
• Final charge on sphere B = (qB + qA / 2) / 2

Example Calculation

For a clearer understanding, let’s consider an example. Suppose:

• qA = +4 μC
• qB = +2 μC

After contact with sphere A:

• Charge on A = 4 μC / 2 = +2 μC
• Charge on C = +2 μC

Now, when C contacts B:

• Total charge = 2 μC + 2 μC = 4 μC
• Charge on B = 4 μC / 2 = +2 μC
• Charge on C = +2 μC

Final Summary

After all interactions, the final charges are:

• Sphere A: +2 μC
• Sphere B: +2 μC

This process illustrates the fundamental principles of charge conservation and the behavior of conductors in electrostatics. Each time the spheres come into contact, they share their total charge equally, leading to a redistribution that ultimately affects their final charges.