Capacitance AP Physics B

Capacitors Consider two separated conductors, like two parallel plates, with external leads to attach to other circuit elements. Such a device is called a capacitor. There is a limit to the amount of charge that a conductor can hold without leaking to the air. There is a certain capacity for holding charge.

Capacitance The capacitance (C) of a conductor is defined as the ratio of the charge (Q) on the conductor to the potential (V) produced. Capacitance:

Capacitance in Farads One farad (F) is the capacitance C of a conductor that holds one coulomb of charge for each volt of potential. Example: When 40  C of charge are placed on a con- ductor, the potential is 8 V. What is the capacitance? C = 5  F

Parallel Plate Capacitance d Area A +Q -Q You will recall from Gauss’ law that E is also: For these two parallel plates: Q is charge on either plate. A is area of plate. And

Example 2. The plates of a parallel plate capacitor have an area of 0.4 m 2 and are 3 mm apart in air. What is the capacitance? 3 mm d A 0.4 m 2 C = 1.18 nF

Applications of Capacitors A Variable Capacitor Changing Area d Changing d Microphone A microphone converts sound waves into an electrical signal (varying voltage) by changing d. The tuner in a radio is a variable capacitor. The changing area A alters capacitance until desired signal is obtained.

Energy of Charged Capacitor The potential energy U of a charged capacitor is equal to the work (qV) required to charge the capacitor. If we consider the average potential difference from 0 to V f to be V/2: Work = Q(V/2) = ½QV

Example 3: In a capacitor, we found its capacitance to be 11.1 nF, the voltage 200 V, and the charge 2.22  C. Find the potential energy U. U = 222  J Verify your answer from the other formulas for P.E. C = 11.1 nF 200 V Q = 2.22  C U = ? Capacitor of Example 3.

Electrical Circuit Symbols Electrical circuits often contain two or more capacitors grouped together and attached to an energy source, such as a battery. The following symbols are often used: + Capacitor Ground Battery - +

Capacitors in Series 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 1 = (4  F)(24 V) = 96  C Q 1 = (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.

Summary: Series Circuits Q = Q 1 = Q 2 = Q 3 V = V 1 + V 2 + V 3 For two capacitors at a time:

Summary: Parallel Circuits Q = Q 1 + Q 2 + Q 3 V = V 1 = V 2 =V 3 For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.

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