Short Version : 23. Electrostatic Energy & Capacitors

23.1. Electrostatic Energy

Electrostatic Energy = work done to assemble the charge configuration of a system.

Reference ( 0 energy): when all component charges are widely separated.

Bringing q1 in place takes no work.

2 2 1 2W q V rBringing in q2 takes

1 1 0 0W q V

12

qq k

a

3 3 1 3 2 3W q V V r rBringing in q3 takes

1 23

q qq k k

a a

Total electrostatic energy1 2 2U W W W 1 2 2 3 3 1

kq q q q q q

a

23.2. Capacitors

Capacitor: pair of conductors carrying equal but opposite charges.

Usage: store electrical energy

Parallel-Plate Capacitor:2 conducting plates of area A separated by a small distance d .

Plates are initially neutral.

They’re charged by connecting to a battery.

Charge transfer plates are equal but oppositely charged.

Large A, small d E 0 outside.

0

ˆinside

E zFar from the edges0

ˆQ

A z ˆinsideV d E z

0

Qd

A

Capacitance

0

QV d

AParallel-plate capacitor: 0 AQ V

d

CV

0 ACd

C = Q / V = capacitance

Parallel-plate capacitor

See Probs 41 & 42

CC

V farad F

Practical capacitor ~ F ( 106 F) or pF ( 1012 F )

0

C d

A

/F m 1dV dQ

C

Charging / Discharging

dQC

dV

Energy Stored in Capacitors

When potential difference between capacitor plates is V,

work required to move charge dQ from to + plate is

Work required to charge the capacitor from 0 to V is

0

VW C V dV 21

2CV = U = energy stored in capacitor

Note:

In a “charged” capacitor, Q is the charge on the + plate.

The total charge of the capacitor is always zero.

plate

plate

dW dQ d

E r V C dV plate platedQ V V E dr < 0

2

2

Q

C 1

2QV

Example 23.1. Parallel-Plate Capacitor

A capacitor consists of two circular metal plates of radius R = 12 cm, separated by

d = 5.0 mm. Find

(a) Its capacitance,

(b) the charge on the plates, and

(c) the stored energy when the capacitor is connected to a 12-V battery.

0

AC

d

100.8 10 F

22

9 3

12 101

4 9 10 / 5.0 10

m

Vm C m

80 pF

(a)

(b)

(c)

Q CV 80 12pF V 960 pC

21

2U CV 21

80 122

pF V 5760 pJ

5.76 nJ

CF

V

Practical Capacitors

Inexpensive capacitors:

Thin plastic sandwiched between aluminum foils

& rolled into cylinder.

Electrolytic capacitors (large capacitance):

Insulating layer developed by electrolysis.

Capacitors in IC circuits (small capacitance):

Alternating conductive & insulating layers.

Dielectrics

Dielectrics: insulators containing molecular dipoles but no free charges.

Dielectric layer lowers V between

capacitor plates by factor 1/ ( > 1).

0

A

d

QC

V

= dielectric constant

Molecular dipoles aligned by E0 .

Dipole fields oppose E0.Net field reduced to E = E0 / .

Hence V = V0 / .Q is unchanged, so C = C0 .

: 2 ~ 10 mostly

Working voltage V = Max safe potential < Ebkd d

Example 23.2. Which Capacitor?

A 100-F capacitor has a working voltage of 20 V,

while a 1.0-F capacitor is rated at 300 V.

Which can store more charge? More energy?

100 FQ CV 6100 10 20F V 32 10 C 2 mC

61 1 10 300FQ F V

43 10 C 0.3mC

2100

1

2FU CV 12 20

2mC V

1

2QV 20 mJ

1

10.3 300

2FU mC V 45mJ

Connecting Capacitors

Two ways to connect 2 electronic components: parallel & series

Parallel: Same V for both components

Q CV 1 2Q Q 1 2C V C V

1 2C C C

Series: Same I (Q) for both components

QV

C 1 2V V

1 2

Q Q

C C

1 2

1 1 1

C C C

1 2C C or C

1 2C C or C

Bursts of Power

San Francisco’s BART train:

KE of deceleration stored as EE in ultracapacitor.

Stored EE is used to accelerate train.

Capacitors deliver higher energy

much more quickly than batteries.

Flash light:

Battery charges capacitor,

which then discharges to give flash.

Other examples:

Defibrillator, controlled nuclear fusion, amusement park rides, hybrid cars, …

23.4. Energy in the Electric Field

Charging a capacitor rearranges charges energy stored in E

Energy density = energy per unit volume

Parallel-plate capacitor:2

2

QU

C

2

02

QAd

E

Uu

A dEnergy density :

2

202

Q

A

2

02

20

1

2E

0

E

is universal2

0

1

2Eu E 3/Eu J m

2

0

1

2 U dVE

Example 23.4. A Thunderstorm

Typical electric fields in thunderstorms average around 105 V/m.

Consider a cylindrical thundercloud with height 10 km and diameter 20 km,

and assume a uniform electric field of 1105 V/m.

Find the electric energy contained in this storm.

111.39 10 J

20

1

2Eu E

EU u V 2 25 3 39 2 2

1 110 / 10 10 10 10

2 4 9 10 /V m m m

Nm C

139 GJ

~ 1400 gallons of gasoline.

Example 23.5. A Shrinking Sphere

A sphere of radius R1 carries charge Q distributed uniformly over its surface.

How much work does it take to compress the sphere to a smaller radius R2 ?

2ˆ

Qkr

E r2

0

1

2 U dVE

1

2

22

1 1

2

R

Rk Q dr

r 1

2

22

2

14

8

R

R

Qk r dr

k r

2

2 1

1 1 1

2k Q

R R

2 10U for R R Work need be done to shrink sphere

2 1U U U 2 1

2 20

14

2 R Rr dr

E

Extra energy stored here