Chapter 20 Capacitors in Series and Parallel

Capacitors in Circuits Like resistors, capacitors in circuits can be connected in series, in parallel, or in more- complex networks containing both series and parallel connections.

Capacitors in Parallel Parallel-connected capacitors all have the same potential difference across their terminals.

Capacitors in Series V1 V2 V3 Capacitors in series all have the same charge, but different potential differences. V1 V2 V3

RC Circuits A capacitor connected in series with a resistor is part of an RC circuit. Resistance limits charging current Capacitance determines ultimate charge

Series Circuits Capacitors or other devices connected along a single path are said to be connected in series. See circuit below: Series connection of capacitors. “+ to – to + …” + - Battery C1 C2 C3 Charge inside dots is induced.

Charge on Capacitors in Series Since inside charge is only induced, the charge on each capacitor is the same. Charge is same: series connection of capacitors. Q = Q1 = Q2 =Q3 Battery C1 C2 C3 + - Q1 Q2 Q3

Voltage on Capacitors in Series Since the potential difference between points A and B is independent of path, the battery voltage V must equal the sum of the voltages across each capacitor. Total voltage V Series connection Sum of voltages V = V1 + V2 + V3 Battery C1 C2 C3 + - V1 V2 V3 • A B

Equivalent Capacitance: Series Q1= Q2 = Q3 + - C1 C2 C3 V1 V2 V3 V = V1 + V2 + V3 Equivalent Ce for capacitors in series:

Example 1. Find the equivalent capacitance of the three capacitors connected in series with a 24-V battery. + - 2 mF C1 C2 C3 24 V 4 mF 6 mF Ce for series: Ce = 1.09 mF

Example 1 (Cont.): The equivalent circuit can be shown as follows with single Ce. + - 2 mF C1 C2 C3 24 V 4 mF 6 mF Ce = 1.09 mF 1.09 mF Ce 24 V Note that the equivalent capacitance Ce for capacitors in series is always less than the least in the circuit. (1.09 mF < 2 mF)

For series circuits: QT = Q1 = Q2 = Q3 Example 1 (Cont.): What is the total charge and the charge on each capacitor? + - 2 mF C1 C2 C3 24 V 4 mF 6 mF 1.09 mF Ce 24 V Ce = 1.09 mF QT = 26.2 mC QT = CeV = (1.09 mF)(24 V); For series circuits: QT = Q1 = Q2 = Q3 Q1 = Q2 = Q3 = 26.2 mC

Example 1 (Cont.): What is the voltage across each capacitor? + - 2 mF C1 C2 C3 24 V 4 mF 6 mF VT = 24 V Note: VT = 13.1 V + 6.55 V + 4.37 V = 24.0 V

Short Cut: Two Series Capacitors The equivalent capacitance Ce for two series capacitors is the product divided by the sum. 3 mF 6 mF + - C1 C2 Example: Ce = 2 mF

Parallel capacitors: “+ to +; - to -” Parallel Circuits Capacitors which are all connected to the same source of potential are said to be connected in parallel. See below: Voltages: VT = V1 = V2 = V3 Parallel capacitors: “+ to +; - to -” C2 C3 C1 + - Charges: QT = Q1 + Q2 + Q3

Equivalent Capacitance: Parallel Parallel capacitors in Parallel: C2 C3 C1 + - Q = Q1 + Q2 + Q3 Equal Voltages: CV = C1V1 + C2V2 + C3V3 Equivalent Ce for capacitors in parallel: Ce = C1 + C2 + C3

Example 2. Find the equivalent capacitance of the three capacitors connected in parallel with a 24-V battery. C2 C3 C1 2 mF 4 mF 6 mF 24 V Q = Q1 + Q2 + Q3 VT = V1 = V2 = V3 Ce for parallel: Ce = (2 + 4 + 6) mF Ce = 12 mF Note that the equivalent capacitance Ce for capacitors in parallel is always greater than the largest in the circuit. (12 mF > 6 mF)

Example 2 (Cont.) Find the total charge QT and charge across each capacitor. 2 mF 4 mF 6 mF 24 V Q = Q1 + Q2 + Q3 Ce = 12 mF V1 = V2 = V3 = 24 V QT = CeV Q1 = (2 mF)(24 V) = 48 mC QT = (12 mF)(24 V) Q2 = (4 mF)(24 V) = 96 mC QT = 288 mC Q3 = (6 mF)(24 V) = 144 mC

Example 3. Find the equivalent capacitance of the circuit drawn below. 4 mF 3 mF 6 mF 24 V C2 C3 Ce = 4 mF + 2 mF Ce = 6 mF C1 4 mF 2 mF 24 V C3,6 Ce 6 mF 24 V

Example 3 (Cont.) Find the total charge QT. Ce = 6 mF C1 4 mF 3 mF 6 mF 24 V C2 C3 Q = CV = (6 mF)(24 V) QT = 144 mC C1 4 mF 2 mF 24 V C3,6 Ce 6 mF

This can also be found from Q = C3,6V3,6 = (2 mF)(24 V) Example 3 (Cont.) Find the charge Q4 and voltage V4 across the the 4-mF capacitor. C1 4 mF 3 mF 6 mF 24 V C2 C3 V4 = VT = 24 V Q4 = (4 mF)(24 V) Q4 = 96 mC The remainder of the charge: (144 mC – 96 mC) is on EACH of the other capacitors. (Series) This can also be found from Q = C3,6V3,6 = (2 mF)(24 V) Q3 = Q6 = 48 mC

Example 3 (Cont.) Find the voltages across the 3 and 6-mF capacitors. Q3 = Q6 = 48 mC C1 4 mF 3 mF 6 mF 24 V C2 C3 Note: V3 + V6 = 16.0 V + 8.00 V = 24 V Use these techniques to find voltage and capacitance across each capacitor in a circuit.