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 Capacitors in series all have the same charge, but different potential differences. V1V1V1V1 V2V2V2V2 V3V3V3V3

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 + …” Charge inside dots is induced. Battery C1C1 C2C2 C3C

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 = Q 1 = Q 2 =Q 3 Battery C1C1 C2C2 C3C Q1Q1 Q2Q2 Q3Q3

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 = V 1 + V 2 + V 3 Battery C1C1 C2C2 C3C V1V1 V2V2 V3V3 AB

Equivalent Capacitance: Series V = V 1 + V 2 + V 3 Q 1 = Q 2 = Q C1C1 C2C2 C3C3 V1V1 V2V2 V3V3 Equivalent C e for capacitors in series:

Example 1. Find the equivalent capacitance of the three capacitors connected in series with a 24-V battery  F C1C1 C2C2 C3C3 24 V 4  F6  F C e for series: C e = 1.09  F

Example 1 (Cont.): The equivalent circuit can be shown as follows with single C e  F C1C1 C2C2 C3C3 24 V 4  F 6  F 1.09  F CeCe 24 V C e = 1.09  F Note that the equivalent capacitance C e for capacitors in series is always less than the least in the circuit. (1.09 < 2 Note that the equivalent capacitance C e for capacitors in series is always less than the least in the circuit. (1.09  F < 2  F)

1.09  F CeCe 24 V  F C1C1 C2C2 C3C3 24 V 4  F 6  F C e = 1.09  F Q T = C e V = (1.09  F)(24 V); Q T  = 26.2  C For series circuits: Q T = Q 1 = Q 2 = Q 3 Q 1 = Q 2 = Q 3 = 26.2  C Example 1 (Cont.): What is the total charge and the charge on each capacitor?

 F C1C1 C2C2 C3C3 24 V 4  F 6  F V T  = 24 V Note: V T = 13.1 V V V = 24.0 V Example 1 (Cont.): What is the voltage across each capacitor?

Short Cut: Two Series Capacitors The equivalent capacitance C e for two series capacitors is the product divided by the sum. 3  F6  F C1C1 C2C2Example: C e = 2  F

Parallel Circuits Capacitors which are all connected to the same source of potential are said to be connected in parallel. See below: Parallel capacitors: “+ to +; - to -” C2C2 C3C3 C1C Charges: Q T = Q 1 + Q 2 + Q 3 Voltages: V T = V 1 = V 2 = V 3

Equivalent Capacitance: Parallel Q = Q 1 + Q 2 + Q 3 Equivalent C e for capacitors in parallel: Equal Voltages: CV = C 1 V 1 + C 2 V 2 + C 3 V 3 Parallel capacitors in Parallel: C2C2 C3C3 C1C C e = C 1 + C 2 + C 3

Example 2. Find the equivalent capacitance of the three capacitors connected in parallel with a 24-V battery. C e for parallel: C e = 12  F C2C2 C3C3 C1C1 2  F4  F6  F 24 V Q = Q 1 + Q 2 + Q 3 V T = V 1 = V 2 = V 3 C e = ( )  F Note that the equivalent capacitance C e for capacitors in parallel is always greater than the largest in the circuit. (12 > 6 Note that the equivalent capacitance C e for capacitors in parallel is always greater than the largest in the circuit. (12  F > 6  F)

Example 2 (Cont.) Find the total charge Q T and charge across each capacitor. C e = 12  F C2C2 C3C3 C1C1 2  F4  F6  F 24 V Q = Q 1 + Q 2 + Q 3 V 1 = V 2 = V 3 = 24 V Q 1 = (2  F)(24 V) = 48  C Q 2 = (4  F)(24 V) = 96  C Q 3 = (6  F)(24 V) = 144  C Q T = C e V Q T = (12  F)(24 V) Q T = 288  C

Example 3. Find the equivalent capacitance of the circuit drawn below. C1C1 4  F 3  F 6  F 24 V C2C2 C3C3 C1C1 4  F 2  F 24 V C 3,6 CeCe 6  F 24 V C e = 4  F + 2  F C e = 6  F

Example 3 (Cont.) Find the total charge Q T. C1C1 4  F 3  F 6  F 24 V C2C2 C3C3 C e = 6  F Q = CV = (6  F)(24 V) Q T = 144  C C1C1 4  F 2  F 24 V C 3,6 CeCe 6  F 24 V

Example 3 (Cont.) Find the charge Q 4 and voltage V 4 across the the 4  F capacitor  C1C1 4  F 3  F 6  F 24 V C2C2 C3C3 V 4 = V T = 24 V Q 4 = (4  F)(24 V) Q 4 = 96  C The remainder of the charge: (144  C – 96  C) is on EACH of the other capacitors. (Series) Q 3 = Q 6 = 48  C This can also be found from Q = C 3,6 V 3,6 = (2  F)(24 V)

Example 3 (Cont.) Find the voltages across the 3 and 6-  F capacitors  C1C1 4  F 3  F 6  F 24 V C2C2 C3C3 Note: V 3 + V 6 = 16.0 V V = 24 V Q 3 = Q 6 = 48  C Use these techniques to find voltage and capacitance across each capacitor in a circuit.