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Chapter 25:Capacitance and Dielectrics

Physics for Scientists and Engineers, 10e

Raymond A. Serway

John W. Jewett, Jr.

Capacitor

Capacitor: combination of two conductors, called plates

Definition of Capacitance

The capacitance C of a capacitor is

defined as the ratio of the magnitude of

the charge on either conductor to the

magnitude of the potential difference

between the conductors:

QC

V

1 F 1 C/V=

Q C V=

Definition of Capacitance

Consider capacitor formed from pair of parallel plates:

• Each plate connected to one terminal of battery

• Acts as source of potential difference

• If capacitor initially uncharged:

• Battery establishes electric field in connecting wires

For plate connected to negative terminal of battery:

• Electric field in wire applies force on electrons in wire immediately

outside this plate:

• Force causes electrons to move onto plate

• Movement continues until plate, wire, and terminal all at same

electric potential

• Once equilibrium attained potential difference no longer exists

between terminal and plate:

• Result: no electric field present in wire and electrons stop

moving

• Plate now carries negative charge

• Similar process occurs at other capacitor plate:

• Electrons move from plate to wire, leaving plate positively

charged

Calculating Capacitance

0 0

QE

A

= =

0

QdV Ed

A = =

0

0

/

Q QC

V Qd A

AC

d

= =

=

Capacitance

0 AC

d

=

The parallel-plate capacitor:

• Proportional to area of plates

• Inversely proportional to plate separation

( )04π

2ln /

LC

b a

=The cylindrical capacitor:

• depends on the radii a and b

• is proportional to the length of the cylinders

An example of this type of geometric arrangement is a coaxial cable, which consists of two concentric

cylindrical conductors separated by an insulator. You probably have a coaxial cable attached to your

television set if you are a subscriber to cable television. The coaxial cable is especially useful for shielding

electrical signals from any possible external influences.

The spherical capacitor:

• depends on the radii a and b ( )04π ab

Cb a

=

−

4πbC a→ =

Combinations of Capacitors

Parallel Combination

1 2V V V = =

tot 1 2

1 1 2 2

Q Q Q

QC

V

C V C V

= +

=

= +

Parallel Combination

tot eqQ C V=

( )

eq 1 2

parallel combination

C C C= +

( )eq 1 2 3+ + parallel combinationC C C C= +

tot 1 1 2 2Q C V C V= +

Series Combination

1 2Q Q Q= =

tot 1 2

1 2

1 2

V V V

Q Q

C C

= +

= +

Series Combination

1 2

eq 1 2

Q QQ

C C C= +

( )eq 1 2

1 1 1 series combination

C C C= +

tot

eq

QV

C =

( )eq 1 2 3

1 1 1 1+ + series combination

C C C C= +

1 2tot

1 2

Q QV

C C = +

Energy Stored in a Charged Capacitor

• Because positive and negative charges separated in

system of two conductors in charged capacitor:

Electric potential energy stored in system

• If plates of charged capacitor connected by conductor

such as wire → Charge moves between each plate

and connecting wire until capacitor uncharged.

Discharge often observed as visible spark

• If you accidentally touch opposite plates of charged capacitor → your fingers act as

pathway for discharge. Result: electric shock – Could be dangerous if high voltages

present (i.e., power supply of home theater system)

• Because charges can be stored in capacitor even when system turned off →

Unplugging system does not make it safe to open case and touch components

inside

Energy Stored in a Charged Capacitor

Energy Stored in a Charged Capacitor

d d dq

W V q qC

= =

2

0 0

1d d

2

Q Qq QW q q q

C C C= = =

( )2

21 1

2 2 2E

QU Q V C V

C= = =

Work necessary to transfer increment of charge dq from

plate carrying charge –q to plate carrying charge q (at

higher electric potential) is:

Total energy:

Energy Stored in a Charged Capacitor

( )21

2EU C V=

( ) ( )2 20

0

1 1

2 2E

AU Ed Ad E

d

= =

For parallel-plate capacitor:

• potential difference related to electric field through V = Ed

• capacitance: C = 0A/d

Energy per unit volume (energy density) is: 2

0

1

2

EE

Uu E

Ad= =

Expression generally valid regardless of source of electric field:

• Energy density in any electric field square of magnitude of

electric field at given point

Portable Defibrillator

• Stored energy released through te heart by conducting electrodes (paddles) which are placed

on both sides of victim’s chest

• Defibrillator can deliver energy to patient in 2 ms

• Paramedics must wait between applications of energy because of time interval necessary for

capacitors to become fully charged

• Capacitors serve as energy reservoirs that can be slowly charged and then quickly discharged

to provide large amounts of energy in a short pulse

• When cardiac fibrillation (random contractions) occurs →

heart produces rapid, irregular pattern of beats

• Fast discharge of energy through heart can return the

organ to normal beat pattern

• Emergency medical teams use portable defibrillators that

contain batteries capable of charging capacitor to high

voltage

• Circuitry actually permits capacitor to be charged to

much higher voltage than that of battery

• Up to 360 J stored in electric field of large

capacitor in defibrillator when fully charged

Capacitors with Dielectrics

0VV

=

0 0

0

0

0

0

/

Q QC

V V

Q

V

C C

= =

=

=

0 AC

d

=

Dielectric: nonconducting material (i.e., rubber, glass, or waxed paper)

𝜀 = dielectric constant of the material

Capacitors with Dielectrics

0 AC

d

=

1) Dielectric provides:

• Increase in capacitance

• Increase in maximum operating voltage

2) If magnitude of electric field 𝐸 = 𝑈/𝑑 in dielectric exceeds dielectric

strength → Insulating properties break down and dielectric begins to

conduct, causing discharge.

3) When selecting capacitor for given application, must consider capacitance

as well as expected voltage across capacitor in circuit.

( )2

21 1

2 2 2E

QU Q V C V

C= = =

Electric Dipole in an Electric Field

2p aq

Electric dipole: two charges of equal magnitude and opposite sign separated by distance a

Electric dipole moment: vector Ԧ𝑝 directed from −q toward +q along line joining

the charges, with magnitude

Electric Dipole in an Electric Field

sinFa =

sin sinaqE pE = =

= τ p E

sinFa + −= =

sinr F rF = =

Energy of Dipole

cosEU pE = −

dW d =

sinpE =

( )

sin sin

cos cos cos

f f f

i i i

f

i

f i

i f

U U d pE d pE d

pE pE

− = = =

= − = −

EU = − p E

Polar Molecules

Microwave cooking

Many molecules (such as those of water) are electric dipoles, meaning that they have

a partial positive charge at one end and a partial negative charge at the other, and

therefore rotate as they try to align themselves with the alternating electric field of

the microwaves. Rotating molecules hit other molecules and put them into motion,

thus dispersing energy. This energy, dispersed as molecular rotations, vibrations

and/or translations in solids and liquids raises the temperature of the food, in a

process similar to heat transfer by contact with a hotter body

Soap and the Dipole Structure of Water

• Grease and oil made up of nonpolar molecules

• Generally not attracted to water

• Plain water not very useful for removing this type of grime

• Soap contains long molecules called surfactants

• In long molecule →

• Polarity characteristics of one end of molecule can be different from those at

other end

• In surfactant molecule:

• One end acts like nonpolar molecule and other acts like polar molecule

• Nonpolar end can attach to grease or oil molecule

• Polar end can attach to water molecule

• Soap serves as chain → linking dirt and water molecules together

• When water rinsed away:

• Grease and oil go with it

Induced Polarization

• Symmetric molecule (figure): no permanent polarization

• Polarization can be induced by placing molecule in electric field.