Three capacitors, of capacitance C1 = 2.0 uF, C2 = 5.0 uF, and C3 = 7.0 uF, are initially charged to 36 V by connecting each, for a few instants, to a 36-V battery. The battery is then removed and the charged capacitors are connected in a closed series circuit, with the positive and negative terminals joined as shown in Figure 7. What will be the final charge on each capacitor ? What will be the voltage across the points PP' ?
The initial charges on each of the three capacitors, q1, q2, and q3, are equal to
After the three capacitors are connected, the charge will redistribute itself. The charges on the three capacitors after the system settles down are equal to Q1, Q2, and Q3. Since charge is a conserved quantity, there is a relation between q1, q2, and q3, and Q1, Q2, and Q3:
The voltage between P and P' can be expressed in terms of C3 and Q3, or in terms of C1, C2, Q1, and Q2:
and
Using eq.(22) the following expressions for Q1 and Q2 can be obtained:
Substituting eq.(25) and eq.(26) into eq.(24) we obtain
Combining eq.(27) and eq.(23), Q3 can be expressed in terms of known variables:
Substituting the known values of the capacitance and initial charges we obtain
The voltage across P and P' can be found by combining eq.(29) and eq.(23):
The charges on capacitor 1 and capacitor 2 are equal to
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