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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

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/d Spherical : 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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