Physics 2102 lecture 12 mon feb
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Physics 2102 Jonathan Dowling. Physics 2102 Lecture: 12 MON FEB. Capacitance II. 25.4–5. V = V AB = V A –V B. C 1. Q 1. V A. V B. A. B. C 2. Q 2. V C. V D. D. C. V = V CD = V C –V D. C eq. Q total.  V=V. Capacitors in Parallel: V=Constant.

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Physics 2102 Lecture: 12 MON FEB

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Physics 2102 lecture 12 mon feb

Physics 2102

Jonathan Dowling

Physics 2102 Lecture: 12 MON FEB

Capacitance II

25.4–5


Capacitors in parallel v constant

V = VAB = VA –VB

C1

Q1

VA

VB

A

B

C2

Q2

VC

VD

D

C

V = VCD = VC –VD

Ceq

Qtotal

V=V

Capacitors in Parallel: V=Constant

  • An ISOLATED wire is an equipotential surface: V=Constant

  • Capacitors in parallel have SAME potential difference but NOT ALWAYS same charge!

  • VAB = VCD = V

  • Qtotal = Q1 + Q2

  • CeqV = C1V + C2V

  • Ceq = C1 + C2

  • Equivalent parallel capacitance = sum of capacitances


Capacitors in series q constant

Q1

Q2

B

C

A

C1

C2

Capacitors in Series: Q=Constant

Isolated Wire:

Q=Q1=Q2=Constant

  • Q1 = Q2 = Q = Constant

  • VAC = VAB + VBC

Q = Q1 = Q2

  • SERIES:

  • Q is same for all capacitors

  • Total potential difference = sum of V

Ceq


Capacitors in parallel and in series

C1

Q1

C2

Q2

Qeq

Q1

Q2

Ceq

C1

C2

Capacitors in parallel and in series

  • In parallel :

    • Cpar = C1 + C2

    • Vpar= V1 = V2

    • Qpar= Q1 + Q2

  • In series :

    1/Cser = 1/C1 + 1/C2

    Vser= V1  + V2

    Qser= Q1 = Q2


Example parallel or series

Parallel: Circuit Splits Cleanly in Two (Constant V)

C1=10 F

C2=20 F

C3=30 F

120V

Example: Parallel or Series?

  • Qi = CiV

  • V = 120V = Constant

  • Q1 = (10 F)(120V) = 1200 C

  • Q2 = (20 F)(120V) = 2400 C

  • Q3 = (30 F)(120V) = 3600 C

What is the charge on each capacitor?

  • Note that:

  • Total charge (7200 mC) is shared between the 3 capacitors in the ratio C1:C2:C3— i.e. 1:2:3


Example parallel or series1

Series: Isolated Islands (Constant Q)

Example: Parallel or Series

C2=20mF

C3=30mF

C1=10mF

What is the potential difference across each capacitor?

  • Q = CserV

  • Q is same for all capacitors

  • Combined Cser is given by:

120V

  • Ceq = 5.46 F (solve above equation)

  • Q = CeqV = (5.46 F)(120V) = 655 C

  • V1= Q/C1 = (655 C)/(10 F) = 65.5 V

  • V2= Q/C2 = (655 C)/(20 F) = 32.75 V

  • V3= Q/C3 = (655 C)/(30 F) = 21.8 V

Note: 120V is shared in the ratio of INVERSE capacitances i.e. (1):(1/2):(1/3)

(largest C gets smallest V)


Example series or parallel

10 F

10F

10F

10V

5F

5F

10V

Example: Series or Parallel?

Neither: Circuit Compilation Needed!

In the circuit shown, what is the charge on the 10F capacitor?

  • The two 5F capacitors are in parallel

  • Replace by 10F

  • Then, we have two 10F capacitors in series

  • So, there is 5V across the 10 F capacitor of interest

  • Hence, Q = (10F )(5V) = 50C


Energy u stored in a capacitor

Energy U Stored in a Capacitor

  • Start out with uncharged capacitor

  • Transfer small amount of charge dq from one plate to the other until charge on each plate has magnitude Q

  • How much work was needed?

dq


Energy stored in electric field of capacitor

General expression for any region with vacuum (or air)

volume = Ad

Energy Stored in Electric Field of Capacitor

  • Energy stored in capacitor: U = Q2/(2C) = CV2/2

  • View the energy as stored in ELECTRIC FIELD

  • For example, parallel plate capacitor: Energy DENSITY = energy/volume = u =


Example

10mF (C1)

20mF (C2)

Example

  • 10mFcapacitor is initially charged to 120V.

  • 20mF capacitor is initially uncharged.

  • Switch is closed, equilibrium is reached.

  • How much energy is dissipated in the process?

Initial charge on 10mF = (10mF)(120V)= 1200mC

After switch is closed, let charges = Q1 and Q2.

Charge is conserved: Q1 + Q2 = 1200mC

  • Q1 = 400mC

  • Q2 = 800mC

  • Vfinal= Q1/C1 = 40 V

Also, Vfinal is same:

Initial energy stored = (1/2)C1Vinitial2 = (0.5)(10mF)(120)2 = 72mJ

Final energy stored = (1/2)(C1 + C2)Vfinal2= (0.5)(30mF)(40)2 = 24mJ

Energy lost (dissipated) = 48mJ


Summary

  • Any two charged conductors form a capacitor.

  • Capacitance : C= Q/V

  • Simple Capacitors:Parallel plates: C = e0 A/dSpherical : C = 4pe0 ab/(b-a)Cylindrical: C = 2pe0 L/ln(b/a)

  • Capacitors in series: same charge, not necessarily equal potential; equivalent capacitance 1/Ceq=1/C1+1/C2+…

  • Capacitors in parallel: same potential; not necessarily same charge; equivalent capacitance Ceq=C1+C2+…

  • Energy in a capacitor: U=Q2/2C=CV2/2; energy densityu=e0E2/2

  • Capacitor with a dielectric: capacitance increasesC’=kC

Summary


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