Solved Problem on Electric Charge
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A sphere A, initially charged with 8 coulombs, is placed in contact with another sphere B charged with −2 coulombs, then they are separated, and sphere B is grounded. Another sphere C, with an initial charge of −5 coulombs, is put in contact with sphere A and then separated. Sphere A is placed again in contact with sphere B and separated. Determine the final charges of each sphere.


Problem data:
  • Charge of sphere A:    QA = 8 C;
  • Charge of sphere B:    QB = −2 C;
  • Charge of sphere C:    QC = −5 C.
Solution

Initially, we have the following situation

Figure 1

When we put spheres A and B in contact, the total charge will be equally distributed by the spheres, and the final charge of each sphere will be the average between the spheres.
\[ \begin{gather} Q_{\small A}=Q_{\small B}=\frac{8\;\mathrm C+(-2\;\mathrm C)}{2}=\frac{8\;\mathrm C-2\;\mathrm C}{2}=\frac{6\;\mathrm C}{2}=3\;\mathrm C \end{gather} \]
Figure 2

When sphere B is grounded, it uncharges, and its charge is equal to zero.
\[ \begin{gather} Q_{\small B}=0 \end{gather} \]
Figure 3

Placing spheres A and C in contact, the total charge will be distribute equally by the spheres, and the final charge of each sphere will be the average between the spheres, and the charge of the sphere is the charge acquired after contact with B ( \( Q_{\small A}=3\;\mathrm C \) )
\[ \begin{gather} Q_{\small A}=Q_{\small C}=\frac{3\;\mathrm C+(-5\;\mathrm C)}{2}=\frac{3\;\mathrm C-5\;\mathrm C}{2}=-\frac{2\;\mathrm C}{2}=-1\;\mathrm C \end{gather} \]
Figure 4

Placing spheres A and B in contact, the total charge will be distribute equally by the spheres, and the final charge of each sphere will be the average between the spheres. For the charge of sphere A, we use the value calculated after contact with C ( \( Q_{\small A}=-1\;\mathrm C \) ), and for sphere B, we use the value after it has been grounded ( \( Q_{\small B}=0 \) )
\[ \begin{gather} Q_{\small A}=Q_{\small B}=\frac{-1\;\mathrm C+0}{2}=-\frac{1\;\mathrm C}{2}=-0.5\;\mathrm C \end{gather} \]
Figure 5

The final charges will be:
  • Charge of the sphere A:  QA = − 0.5 C;
  • Charge of the sphere B:  QB = − 0.5 C;
  • Charge of the sphere C:  QC = − 1 C.
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