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Chapter 26B - Capacitor Circuits A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007

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Objectives: After completing this module, you should be able to: Calculate the equivalent capacitance of a number of capacitors connected in series or in parallel. Determine the charge and voltage across any chosen capacitor in a network when given capacitances and the externally applied potential difference.

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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 - +

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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 C3C3 + + - - + + + + - - - -

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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 C3C3 + + - - + + + + - - - - Q1Q1 Q2Q2 Q3Q3

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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 C3C3 + + - - + + + + - - - - V1V1 V2V2 V3V3 AB

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

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Example 1. Find the equivalent capacitance of the three capacitors connected in series with a 24-V battery. + + - - + + + + - - - - 2 F C1C1 C2C2 C3C3 24 V 4 F6 F C e for series: C e = 1.09 F

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Example 1 (Cont.): The equivalent circuit can be shown as follows with single C e. + + - - + + + + - - - - 2 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)

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1.09 F CeCe 24 V + + - - + + + + - - - - 2 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?

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+ + - - + + + + - - - - 2 F C1C1 C2C2 C3C3 24 V 4 F 6 F V T = 24 V Note: V T = 13.1 V + 6.55 V + 4.37 V = 24.0 V Example 1 (Cont.): What is the voltage across each capacitor?

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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

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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 C1C1 ++ -- ++ -- ++ -- Charges: Q T = Q 1 + Q 2 + Q 3 Voltages: V T = V 1 = V 2 = V 3

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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 C1C1 ++ -- ++ -- ++ -- C e = C 1 + C 2 + C 3

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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 = (2 + 4 + 6) 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)

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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

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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

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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

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Example 3 (Cont.) Find the charge Q 4 and voltage V 4 across the the 4 F capacitor 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)

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Example 3 (Cont.) Find the voltages across the 3 and 6- F capacitors 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 + 8.00 V = 24 V Q 3 = Q 6 = 48 C Use these techniques to find voltage and capacitance across each capacitor in a circuit.

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

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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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CONCLUSION: Chapter 26B Capacitor Circuits

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