Solved Problem on Electric Charge

We have three identical conducting spheres insulated from each other. The electric charge on sphere A is equal to 12 μC, sphere B has a charge equal to 9 μC, and sphere C has a charge equal to −3 μC. The following operations are performed with these spheres:
1 – The three spheres are placed in contact simultaneously for a certain time and then separated;
2 – Sphere B is connected to the ground and then isolated;
3 – Spheres B and C are placed in contact and separated;
4 – Spheres A and C are placed in contact and separated.
Determine, in these conditions, the final charges of the spheres.

Problem data:

Sphere A charge: Q_A = 12 μC;
Sphere B charge: Q_B = 9 μC;
Sphere C charge: Q_C = −3 μC.

Solution

Initially, we have the following situation

Figure 1

In operation (1), we place spheres A, B, and C in contact, the total charge will be distributed equally across the spheres, and the final charge of each sphere will be the average of the charges on the spheres (Figure 2)

\[ \begin{gather} Q_{A}=Q_{B}=Q_{C}=\frac{Q_{A}+Q_{B}+Q_{C}}{3}\\ Q_{A}=Q_{B}=Q_{C}=\frac{12\times 10^{-6}+9\times 10^{-6}+(-3\times 10^{-6})}{3}\\ Q_{A}=Q_{B}=Q_{C}=\frac{18\times 10^{-6}}{3}\\ Q_{A}=Q_{B}=Q_{C}=6\times 10^{-6}=6\;\mu\text{C} \end{gather} \]

Figure 2

In operation (2), the sphere B is connected to the ground, it is discharged and its charge is equal to zero (Figure 3)

\[ Q_{B}=0 \]

Figure 3

In operation (3), spheres B and C are brought into contact, and the charges of the two spheres will be equally distributed over them, but now the initial charge of B will be the value found above, Q_B = 0, (Figure 4)

\[ \begin{gather} Q_{B}=Q_{C}=\frac{Q_{B}+Q_{C}}{2}\\ Q_{B}=Q_{C}=\frac{0+6\times 10^{-6}}{2}\\ Q_{B}=Q_{C}=\frac{6\times 10^{-6}}{2}\\ Q_{B}=Q_{C}=3\times 10^{-6}=3\;\mu\text{C} \end{gather} \]

Figure 4

In operation (4), spheres A and C are brought into contact, and the charges of the two spheres will be equally distributed over them, but now the initial charge of C will be the value found above, Q_C = 3 μC, (Figure 5)

\[ \begin{gather} Q_{A}=Q_{B}=\frac{Q_{A}+Q_{B}}{2}\\ Q_{A}=Q_{B}=\frac{6\times 10^{-6}+3\times 10^{-6}}{2}\\ Q_{A}=Q_{B}=\frac{9\times 10^{-6}}{2}\\ Q_{A}=Q_{B}=4,5\times 10^{-6}=4,5\;\mu\text{C} \end{gather} \]

Figure 5

The electric charges values will be:
Sphere A: Q_A= 4,5 μC;
Sphere B: Q_B= 3 μC;
Sphere C: Q_C= 4,5 μC.

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