Physics 2113 Lecture: 17 WED 25 FEB

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Physics 2113 Lecture 17 WED 25 FEB

• Capacitance II

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Capacitors in Parallel VConstant

• An ISOLATED wire is an equipotential surface
  VConstant
• 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

PAR-V (Parallel V the Same)
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Capacitors in Series QConstant
Isolated Wire QQ1Q2Constant

• Q1 Q2 Q Constant
• VAC VAB VBC

SERI-Q Series Q the Same
Q Q1 Q2

• SERIES
• Q is same for all capacitors
• Total potential difference sum of V

Ceq
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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

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Example Parallel or Series?

• What is the charge on each capacitor?

• Qi CiV
• V 120V on ALL Capacitors (PAR-V)
• Q1 (10 µF)(120V) 1200 µC
• Q2 (20 µF)(120V) 2400 µC
• Q3 (30 µF)(120V) 3600 µC

• Note that
• Total charge (7200 µC) is shared between the 3
  capacitors in the ratio C1C2C3 i.e. 123

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Example Parallel or Series

• What is the potential difference across each
  capacitor?

C330mF
C220mF
C110mF

• Q CserV
• Q is same for all capacitors (SERI-Q)
• Combined Cser is given by

120V

• Ceq 5.46 µF (solve above equation)
• Q CeqV (5.46 µF)(120V) 655 µC
• V1 Q/C1 (655 µC)/(10 µF) 65.5 V
• V2 Q/C2 (655 µC)/(20 µF) 32.75 V
• V3 Q/C3 (655 µC)/(30 µF) 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)
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Example Series or Parallel?
Neither Circuit Simplification Needed!

• In the circuit shown, what is the charge on the
  10µF capacitor?

• The two 5µF capacitors are in parallel
• Replace by 10µF
• Then, we have two 10µF capacitors in series
• So, there is 5V across the 10 µF capacitor of
  interest by symmetry
• Hence, Q (10µF )(5V) 50µC

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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
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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

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Dielectric Constant

• If the space between capacitor plates is filled
  by a dielectric, the capacitance INCREASES by a
  factor ?
• This is a useful, working definition for
  dielectric constant.
• Typical values of ? are 10200 but it is always
  greater than 1!

The ? and the constant e?eo are both called
dielectric constants. The ? has no units
(dimensionless). Trick Just substitute e?eo for
eo in all the previous formulas!
C ?e0 A/d
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Atomic View
Emol
Molecules set up counter E field Emol that
somewhat cancels out capacitor field Ecap. This
avoids sparking (dielectric breakdown) by keeping
field inside dielectric small. Hence the bigger
the dielectric constant the more charge you can
store on the capacitor.
Ecap
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Example Battery Connected Voltage V is
Constant but Charge Q Changes

• Capacitor has charge Q, voltage V
• Battery remains connected while dielectric slab
  is inserted.
• Do the following increase, decrease or stay the
  same
• Potential difference?
• Capacitance?
• Charge?
• Electric field?

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Example Battery Connected Voltage V is
Constant but Charge Q Changes

• Initial values
  capacitance C charge Q potential difference
  V electric field E
• Battery remains connected
• V is FIXED Vnew V (same)
• Cnew ?C (increases)
• Qnew (?C)V ?Q (increases).
• Since Vnew V, Enew V/dE (same)

Energy stored? ue0E2/2 gt u?e0E2/2
eE2/2 increases
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Example Battery Disconnected Voltage V
Changes but Charge Q is Constant

• Capacitor has charge Q, voltage V
• Battery remains is disconnected then dielectric
  slab is inserted.
• Do the following increase, decrease or stay the
  same
• Potential difference?
• Capacitance?
• Charge?
• Electric field?

dielectric slab
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Example Battery Disconnected Voltage V
Changes but Charge Q is Constant

• Initial values
  capacitance C charge Q potential difference
  V electric field E
• Battery remains disconnected
• Q is FIXED Qnew Q (same)
• Cnew ?C (increases)
• Vnew Q/Cnew Q/(?C) (decreases).
• Since Vnew lt V, Enew Vnew/d E/? (decreases)

dielectric slab
Energy stored?
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Summary

• Any two charged conductors form a capacitor.
• Capacitance C Q/V
• Simple Capacitors Parallel plates C e0
  A/d Spherical C 4p e0 ab/(b-a) Cylindrical
  C 2p e0 L/ln(b/a)
• Capacitors in series same charge, not
  necessarily equal potential equivalent
  capacitance 1/Ceq1/C11/C2
• Capacitors in parallel same potential not
  necessarily same charge equivalent capacitance
  CeqC1C2
• Energy in a capacitor UQ2/2CCV2/2 energy
  density ue0E2/2
• Capacitor with a dielectric capacitance
  increases C?C