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1 University of Rhode Island PHY 204: Elementary Physics II Physics Course Materials Capacitors I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: Abstract Part seven of course materials for Elementary Physics II (PHY 204), taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Some of the slides contain figures from the textbook, Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, 5 th /6 th editions. The copyright to these figures is owned by W.H. Freeman. We acknowledge permission from W.H. Freeman to use them on this course web page. The textbook figures are not to be used or copied for any purpose outside this class without direct permission from W.H. Freeman. Recommended Citation Müller, Gerhard, "07. Capacitors I" (2015). PHY 204: Elementary Physics II. Paper This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in PHY 204: Elementary Physics II by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.

2 Capacitor and Capacitance Capacitor (device): Two oppositely charged conductors separated by an insulator. The charges +Q and Q on conductors generate an electric field E and a potential difference V (voltage). Only one conductor may be present. Then the relevant potential difference is between the conductor and a point at infinity. Capacitance (device property): Definition: C = Q V SI unit: 1F = 1C/V (one Farad) 14/9/2015 [tsl103 1/21]

3 Parallel-Plate Capacitor A: area of each plate d: distance between plates Q: magnitude of charge on inside surface of each plate Charge per unit area (magnitude) on each plate: σ = Q A Uniform electric field between plates: E = σ ǫ 0 = Q ǫ 0 A Voltage between plates: V V + V = Ed = Qd ǫ 0 A Capacitance for parallel-plate geometry: C Q V = ǫ 0A d +Q Q E V+ V d 14/9/2015 [tsl104 2/21]

4 Cylindrical Capacitor Conducting cylinder of radius a and length L surrounded concentrically by conducting cylindrical shell of inner radius b and equal length. Assumption: L b. λ: charge per unit length (magnitude) on each cylinder Q = λl: magnitude of charge on each cylinder Electric field between cylinders: use Gauss law E[2πrL] = λl E(r) = λ ǫ 0 2πǫ 0 r Electric potential between cylinders: use V (a) = 0 Z r V (r) = E(r)dr = λ Z r dr 2πǫ 0 r = λ ln r 2πǫ 0 a a Voltage between cylinders: V V + V = V (a) V (b) = Capacitance for cylindrical geometry: C Q V = 2πǫ 0L ln(b/a) a Q 2πǫ 0 L ln b a Q +Q a E b 14/9/2015 [tsl105 3/21]

5 Spherical Capacitor Conducting sphere of radius a surrounded concentrically by conducting spherical shell of inner radius b. Q: magnitude of charge on each sphere Electric field between spheres: use Gauss law E[4πr 2 ] = Q E(r) = Q ǫ 0 4πǫ 0 r 2 Electric potential between spheres: use V (a) = 0 Z r V (r) = E(r)dr = Q Z r dr 4πǫ 0 r 2 = Q» 1 4πǫ 0 r 1 a Voltage between spheres: a V V + V = V (a) V (b) = Capacitance for spherical geometry: C Q V = 4πǫ ab 0 b a a Q b a 4πǫ 0 ab Q +Q a E b 14/9/2015 [tsl106 4/21]

6 Energy Stored in Capacitor Charging a capacitor requires work. The work done is equal to the potential energy stored in the capacitor. While charging, V increases linearly with q: V (q) = q C. Increment of potential energy: du = V dq = q C dq. Potential energy of charged capacitor: U = Z Q 0 V dq = 1 C Z Q 0 qdq = Q2 2C = 1 2 CV 2 = 1 2 QV. Q: where is the potential energy stored? A: in the electric field. 14/9/2015 [tsl107 5/21]

7 Energy Density Between Parallel Plates Energy is stored in the electric field between the plates of a capacitor. Capacitance: C = ǫ 0A d. Voltage: V = Ed. Potential energy: U = 1 2 CV 2 = 1 2 ǫ 0E 2 (Ad). +Q Q Volume between the plates: Ad. Energy density of the electric field: u E = U Ad = 1 2 ǫ 0E 2 E A A d 14/9/2015 [tsl108 6/21]

8 Integrating Energy Density in Spherical Capacitor Electric field: E(r) = Voltage: V = Q b a 4πǫ 0 ab Q 1 4πǫ 0 r 2 = Q 4πǫ 0 Energy density: u E (r) = 1 2 ǫ 0E 2 (r)» 1 a 1 b Q +Q a E b Energy stored in capacitor: U = U = U = 1 2 Z b a 1 2 ǫ 0 Q 2 4πǫ 0 Q 2 Z b a 1 (4πǫ 0 ) 2 r 4 (4πr2 )dr Z b a 1 r 2 dr = 1 2 Q 2 4πǫ 0 u E (r)(4πr 2 )dr» 1 a 1 b = 1 2 QV 14/9/2015 [tsl109 7/21]

9 Capacitor Problem (1) Consider two oppositely charged parallel plates separated by a very small distance d. What happens when the plates are pulled apart a fraction of d? Will the quantities listed below increase or decrease in magnitude or stay unchanged? (a) Electric field E between the plates. (b) Voltage V across the plates. (c) Capacitance C of the device. (d) Energy U stored in the device Ε d 14/9/2015 [tsl110 8/21]

10 Capacitors Connected in Parallel Find the equivalent capacitance of two capacitors connected in parallel: Charge on capacitors: Q 1 + Q 2 = Q Voltage across capacitors: V 1 = V 2 = V Equivalent capacitance: C Q V = Q 1 + Q 2 V C = C 1 + C 2 = Q 1 V 1 + Q 2 V 2 V 0 V 1 = V x C1 Q +Q1 1 V0 V 0 + V +Q Q 2 2 C 2 V 0 V = V 2 x 14/9/2015 [tsl111 9/21]

11 Capacitors Connected in Series Find the equivalent capacitance of two capacitors connected in series: Charge on capacitors: Q 1 = Q 2 = Q Voltage across capacitors: V 1 + V 2 = V Equivalent capacitance: 1 C V Q = V 1 + V 2 Q = V 1 Q 1 + V 2 Q 2 1 C = 1 C C 2 V 0 C 1 V 1 V 0 Q +Q Q C 2. V 2 x +Q V 0 + V 14/9/2015 [tsl112 10/21]

12 Capacitor Problem (2) Consider two equal capacitors connected in series. (a) Find the voltages V A V B, V B V C, V A V D. (b) Find the charge Q A on plate A. (c) Find the electric field E between plates C and D. 2cm 2cm 24V B D 12V A 2µF. C 2µF 14/9/2015 [tsl113 11/21]

13 Capacitor Circuit (1) Find the equivalent capacitances of the two capacitor networks. All capacitors have a capacitance of 1µF. (a) (b) 14/9/2015 [tsl114 12/21]

14 Capacitor Circuit (2) Consider the two capacitors connected in parallel. (a) Which capacitor has the higher voltage? (b) Which capacitor has more charge? (c) Which capacitor has more energy? 12V C = 1 1 µf C = 2 2 µf Consider the two capacitors connected in series. (d) Which capacitor has the higher voltage? (e) Which capacitor has more charge? (f) Which capacitor has more energy? C = 1 1 µf C = 2 2 µf 12V 14/9/2015 [tsl115 13/21]

15 Capacitor Circuit (3) Connect the three capacitors in such a way that the equivalent capacitance is C eq = 4µF. Draw the circuit diagram. 2µF 2µF 3µF 4µ F 14/9/2015 [tsl116 14/21]

16 Capacitor Circuit (4) Connect the three capacitors in such a way that the equivalent capacitance is C eq = 2µF. Draw the circuit diagram. 1µF 3µF 5µF 2µF 14/9/2015 [tsl117 15/21]

17 Capacitor Circuit (5) Find the equivalent capacitances of the following circuits. 2µF 1µF (a) 3µF 1µF 2µF 1µF (b) 3µ F 1µF 14/9/2015 [tsl118 16/21]

18 Capacitor Circuit (6) (a) Name two capacitors from the circuit on the left that are connected in series. (b) Name two capacitors from the circuit on the right that are connected in parallel. C 1 C 1 C 2 C 3 C 4 C 2 C 3 C 4 C 5 C 6 14/9/2015 [tsl119 17/21]

19 Capacitor Circuit (7) (a) In the circuit shown the switch is first thrown to A. Find the charge Q 0 and the energy U A on the capacitor C 1 once it is charged up. (b) Then the switch is thrown to B, which charges up the capacitors C 2 and C 3. The capacitor C 1 is partially discharged in the process. Find the charges Q 1, Q 2, Q 3 on all three capacitors and the voltages V 1, V 2, V 3 across each capacitor once equilibrium has been reached again. What is the energy U B now stored in the circuit? A B 0 S V = 120V C = 4 1 µf C = 3µF 2 C = 6 3 µf 14/9/2015 [tsl120 18/21]

20 Capacitor Circuit (8) In the circuit shown find the charges Q 1, Q 2, Q 3, Q 4 on each capacitor and the voltages V 1, V 2, V 3, V 4 across each capacitor (a) when the switch S is open, (b) when the switch S is closed. C = 1 1 µf C = 3 3 µf S C 2 = 2µF C 4 = 4 µf V = 12V 14/9/2015 [tsl122 19/21]

21 Capacitor Circuit (9) The circuit of capacitors connected to a battery is at equilibrium. (a) Find the equivalent capacitance C eq. (b) Find the total energy U stored in the circuit (excluding the battery). (c) Find the the charge Q 3 on capacitor C 3. (d) Find the voltage V 2 across capacitor C 2. C 1 = 3µ F 12V C3 = 4µF C 2 = 6 µ F 14/9/2015 [tsl335 20/21]

22 More Complex Capacitor Circuit No two capacitors are in parallel or in series. Solution requires different strategy: zero charge on each conductor (here color coded), zero voltage around any closed loop. Specifications: C 1,..., Q 5, V. Five equations for unknowns Q 1,..., Q 5 : Q 1 + Q 2 Q 4 Q 5 = 0 Q 3 + Q 4 Q 1 = 0 Q 5 C 5 + Q 3 C 3 Q 4 C 4 = 0 Q 2 C 2 Q 1 C 1 Q 3 C 3 = 0 V Q 1 C 1 Q 4 C 4 = 0 Equivalent capacitance: C eq = Q 1 + Q 2 V + + C 1 + C 4 C 2 C 3 V + + C 5 (a) C m = 1pF, m = 1,..., 5 and V = 1V: C eq = 1pF, Q 3 = 0, Q 1 = Q 2 = Q 4 = Q 5 = 1 2 pc. (b) C m = m pf, m = 1,..., 5 and V = 1V: C eq = pf, Q 1 = pc, Q 2 = pc, Q 3 = 9 71 pc, Q 4 = pc, Q 5 = pc. 14/9/2015 [tsl511 21/21]