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Chapter 17Chapter 17

Capacitance and Capacitors!

• C = q / V V= voltage q = charge

• Capacitors in parallel• Capacitors in series

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Capacitors, Batteries, & Circuits

Capacitors, Batteries, & Circuits

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Single Capacitor Example

Single Capacitor Example

Voltage = 12 voltsCapacitance = 20 microfarads

Charge = ?

q = C * V = 20 e-6 f * 12 V = 240 microcoulombs

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Example: Capacitors in Parallel

Example: Capacitors in Parallel

A circuit consists of a 6.0 Fcapacitor, a 12.0 F capacitor, a 12 Vbattery, and a switch. Initially, theswitch is open and the capacitors discharged.The switch is then closed, and thecapacitors charge. When the capacitors are fully chargedand the open-circuit terminal voltage of thebattery is restored:(a) What is potential of each conductor inthe circuit?(b) What is the charge on each capacitor plate?(c) What total charge has passed through the battery?

12 V; 0 V.a bV V 6 6

1 1 (6.0 10 F)(12 V) 72 10 C 72 CaQ CV 6 6

2 2 (12.0 10 F)(12 V) 144 10 C 144 CaQ C V

total 1 2 (72 C) (144 C) 216 CQ Q Q

total /

(216 C) / (12 V)

18 F

eff aC Q V

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Capacitors in ParallelCapacitors in Parallel

The connected plates are at the same potential, so they form an effective capacitor with a plate area that is the sum of the two plate areas. Since C~A, Ceff = C1+C2, i.e., capacitors in parallel add capacitances.

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Example: Capacitors in Series

Example: Capacitors in Series

A circuit consists of a 6.0 F capacitor, a 12.0 Fcapacitor, a 12 V battery, and a switch. Initially, theswitch is open and the capacitors discharged. Theswitch is then closed, and the capacitors charge. When the capacitors are fully charged and theopen-circuit terminal voltage of the battery is restored:(a) What is potential of each conductor in the circuit?(b) What is the charge on each capacitor plate?(c) What total charge has passed through the battery?

12 V; 0 V; ?a b mV V V 1 2; .a m m bV V V V V V

1 1 1 1 a mQ CV C V V 2 2 2 2 m bQ C V C V V

1 2 1 2a m m bQ Q C V V C V V

1 2

1 2 1 2

1 212 V 0 V 4 V

3 3m a b

C CV V V

C C C C

1 2 1Q (6.0 F) (12 V) (4 V) 48 Ctotal a mQ Q C V V

total /

(48 C) / (12 V)

4 F

eff aC Q V

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Capacitors in SeriesCapacitors in Series The connected string of capacitors all have the same charge Q on their plates, and the potential drops across the component capacitors add.

1 1 2 2 1 2( )effQ CV C V C V V

2 1 1 2( / )V V C C

1 1 1 1

1 2 1 1 2

1 2

1 2 1 2

1 ( / )

1

1/ 1/

eff

CV CVC

V V V C C

C C

C C C C

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Example:Using the C Equivalence

Formula

Example:Using the C Equivalence

Formula A 6.0 F capacitor and a 12.0 F capacitor, each initially uncharged, are connected in series across a 12 V battery.

(a) Find the charge Q on each capacitor.

(b) Find the potential difference across each capacitor.

1 2

1 14.0 F

1/ 1/ 1/ (6.0 F) 1/ (12.0 F)effC C C

(4.0 F)(12 ) 48 CeffQ C V V

1 1/ (48 C) / (6.0 F) 8.0 V Q C V

2 2/ (48 C) / (12.0 F) 4.0 V Q C V

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Dielectric MaterialsDielectric Materials

E off E on

There is a class of polarizable dielectric materials that have an important application in the construction of capacitors. In an electric field their tiny dipoles line up, reducing the E field and potential difference and increasing the capacitance:

0

C

AQC

V d

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Electric Fields and Dielectrics

Electric Fields and Dielectrics

In an external field EO, neutral molecules can polarize. The induced electric field E’ produced by the dipoles will be in the opposite direction from the external field EO. Therefore, in the interior of the slab the resulting field is E = EO-E’.

The polarization of the material has the net effect of producing a sheet of positive charge on the right surface and a sheet of negative charge on the left surface, with E’ being the field made by these sheets of charge.

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Capacitors and Dielectrics

Capacitors and Dielectrics

If a capacitor connected to a battery, so that it has a charge q, and then a dielectric material of dielectric constant e is placed in the gap, the potential is unchanged but the charge becomes eq.

If a capacitor is given a charge q, and then a dielectric material of dielectric constant e is placed in the gap, the charge q is unchanged, but the potential drops to V/e.

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Example: A Dielectric in a Parallel-Plate Capacitor

Example: A Dielectric in a Parallel-Plate Capacitor

A parallel plate capacitor has square plates of edge-length 10 cm and a gap of 4.0 mm. A dielectric slab with = 2.0 has dimensions 10 cm x 10 cm x 4 mm.(a) What is the capacitance without the dielectric?(b) What is the capacitance with the dielectric in the gap?(c) What is the capacitance if a similar dielectric with a thickness of 3.0 mm is placed in the gap?

1 20.0040 m; 0.0030 m; 0.0010 md d d 2

1 0 / (8.85 pF/m)(0.10 m) / (0.0040 m) 22.1 pFC A d

2 0 1/ (2.0)(22.1 pF) 44.2 pFC A d C

1 1

3 1 0 2 0 1 1 1

2/ / 3 / 4 1/ 4 4 35.4 pF

5C d A d A C C C

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Energy Storage with Dielectrics

Energy Storage with Dielectrics

1 1 1 1 12 2 2 20 002 2 2 2 2

( ) ( ) ( )A A

U QV CV Ed Ed E Add d

1 1 12 2 20 0 02 2 2

1E

Uu E E E

Ad

ElectricField

MechanicalStress onDielectric

A

d

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Molecular View of Dielectrics

Molecular View of Dielectrics

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Dielectrics and E FieldsDielectrics and E Fields