Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of -2q. Sphere B carries a charge of +9q. Sphere C carries no net charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Last, sphere C is touched to sphere B and separated from it.
(a) How much charge ends up on sphere C?
(b) What is the total charge on the three spheres before they are allowed to touch each other?
(c) What is the total charge on the three spheres after they have touched?
(a) To determine the charge on sphere C, we need to analyze the transfer of charges during each step.
1. Sphere A (-2q) and Sphere B (+9q) are touched together:
- When two conducting objects touch, they will try to reach equilibrium by redistributing charge.
- Since sphere A has excess negative charge (-2q) and sphere B has excess positive charge (+9q), some charge will transfer between them to neutralize both objects.
- To find the new charge on each sphere, we can calculate the average charge and assign it to both spheres.
- The average charge is (2q + 9q) / 2 = 11q / 2 = 5.5q.
- So, after touching, both sphere A and sphere B will have a charge of +5.5q.
2. Sphere C is touched to sphere A:
- Since they touch, charge can transfer between them.
- Since sphere A has a charge of +5.5q and sphere C initially has no net charge, some charge will transfer between them to neutralize both objects.
- In this case, we will assume sphere C acquires a charge of -5.5q to neutralize the charge of sphere A.
3. Sphere C is touched to sphere B:
- Since they touch, charge can transfer between them.
- At this point, sphere B has a charge of +5.5q (from the previous step) and sphere C has a charge of -5.5q (from the previous step).
- To neutralize both objects, some charge will transfer between them.
- Since the magnitudes of their charges are the same, they will completely neutralize each other.
- Thus, the charge on sphere C after this step will be 0q.
Therefore, the final charge on sphere C is 0q.
(b) Before they are allowed to touch each other, sphere A has a charge of -2q, sphere B has a charge of +9q, and sphere C has no net charge.
(c) After they have touched, the total charge on the three spheres can be found by adding up their individual charges:
Total charge = Charge on sphere A + Charge on sphere B + Charge on sphere C
Total charge = +5.5q + (+5.5q) + 0q
Total charge = 11q
Therefore, the total charge on the three spheres after they have touched is 11q.
To solve this problem, let's analyze each step step by step.
(a) To determine the charge on sphere C, we need to consider the transfer of charges during each step.
Step 1: Spheres A and B are touched together and then separated.
When two spheres touch each other, charge can be transferred between them until they reach equal potentials. Since sphere A has a charge of -2q and sphere B has a charge of +9q, a net charge of +7q is transferred from sphere B to sphere A. After separation, sphere A will have a charge of -2q + 7q = +5q, while sphere B will have a charge of +9q - 7q = +2q.
Step 2: Sphere C is then touched to sphere A and separated from it.
Since sphere C carries no net charge initially, when it is touched to sphere A, charge can be transferred until they reach equal potentials. Since sphere A has a charge of +5q, the net charge of +5q is transferred from sphere A to sphere C. After separation, sphere A will have a charge of 0 (no charge transferred), while sphere C will have a charge of +5q.
Step 3: Sphere C is touched to sphere B and separated from it.
Similarly, when sphere C is touched to sphere B, charge is transferred until they reach equal potentials. Since sphere B has a charge of +2q, the net charge of +2q is transferred from sphere B to sphere C. After separation, sphere B will have a charge of 0 (no charge transferred), while sphere C will have a charge of +5q + 2q = +7q.
Therefore, the final charge on sphere C is +7q.
(b) To determine the total charge on the three spheres before they touch each other, we simply add up their individual charges.
Total charge = charge on sphere A + charge on sphere B + charge on sphere C
Total charge = -2q + 9q + 0 = 7q
Therefore, the total charge on the three spheres before they are allowed to touch each other is 7q.
(c) To determine the total charge on the three spheres after they have touched, we consider the charge redistribution during each step.
After touching each other in Step 1:
Charge on sphere A = +5q
Charge on sphere B = +2q
Charge on sphere C = 0
After touching sphere A in Step 2:
Charge on sphere A = 0
Charge on sphere B = +2q
Charge on sphere C = +5q
After touching sphere B in Step 3:
Charge on sphere A = 0
Charge on sphere B = 0
Charge on sphere C = +7q
Therefore, the total charge on the three spheres after they have touched is +7q.