Capacitors and capacitance

Capacitor is a sort of storage

Capacitance is measured in Farads

Each of the plates of a capacitor is metallic; each of the plates is equipotential

Parallel-plate capacitorConsider two large metal plates which are parallel to each other and separated by a distance d small compared with their width.

σ+σ−

Area A+ + + + + + + +− − − − − − − −

The field between plates is

VΔd

y

AQd

AdAddy

dyEbottomVtopV

d

d

y

0000 0

0

)(

)]()([

εεσ

εσ

εσ

−==−=−

==−−

∫

∫

0 0y

QEA

σε ε

= − = −

the capacitance is the ratio VQ Δ/

Capacitance is measured in farads (in honor of Faraday)

A/d is in meters, therefore, farad can be related to other units. How to reconcile farad with other units?

If charge (Q) is in coulombs, V (voltage) is in volts, the capacitance (C) is in farads

1 1 1C/V 1 /F farad coulomb volt≡ = =

0~QC LV

ε=Δ

How to reconcile farad with other units? 1F=1C/V=1C2/(Nm)=1C2/J

9 2 2 90

120

1/(4 ) 9×10 N×m /C 9×10 m/F

8.85 10 F/m

πε

ε −

= =

= ×

Units:electric field is measured in

N/Celectric potential is thus measured in

(N/C)m=V

electric field is, therefore, measured in:V/m=N/C

1volt=1 (N/C)m=1 J/C=1 joule/coulomb1V=1 (N/C)m

dA

VQC 0ε=Δ

=

If charge (Q) is in coulombs, V (voltage) is in volts, the capacitance (C) is in farads

Similarly to 1 Coulomb, 1 Farad is absolutely unrealistic (too large) quantity.

Capacitance in real life is measured from picofarads to microfarads:

120 8.85 10 /F mε −= ×

12 61 10 1 10pF F F Fμ− −= =

mmmd 3101 −==

Capacitors and capacitance

00~Q AC L

V dε ε= =

Δ

Spherical capacitor

20 0 0

0

02

0 0

( )4 4 4

4

( ) 4

4( )

b b

a a

abR R

aba bR R

a b

b a

b a

a b b ab a

QCV

Q Q QV drE r drr R R

R RCR R

a R C R

R Ab R R R R d CR R d

πε πε πε

πε

πε

πε ε

=

= = = −

=−

→ ∞ =

≈ >> − = = =−

∫ ∫

0~C Lε

parallel-plate capacitor

capacitance of a sphere

Capacitance of a human body

Farad is a too big unit. Capacitance in real life is measured from

picofarads, 10-12F , up to microfarads, 10-6F.

Capacitance of a human body is comparable with those used in radio tuning.

Capacitance used in cameras as a storage of energy for a flash light is of a few hundred microfarads.

120 0

2

~ 4 8.85 10 /

~ 10

C L F m

C pF

πε ε −= ×12 610 10pF F F Fμ− −= =

Commercial capacitors

Coaxial cable

( ) 0 ???V ∞ =

0 0

[ ( ) ( )]

ln | ???2 2

r

rr

V V r Edr

dy yy

λ λπε πε

∞

∞∞

− ∞ − = =

= =

∫

∫

r

rr r

0( ) 0V r = 0 0

0

0 0

0

0

( )] ln |2 2

ln2

r rrr

r r

dyV r Edr yy

rr

λ λπε πε

λπε

= = =

=

∫ ∫r

rr r

only potential difference matters, measure potential difference with respect to

an arbitrary point r0

0 0

0

( )] ln |2 2

ln2

b b

b

a

a a

r rrr

r r

b

a

dyV r Edr yy

rr

λ λπε πε

λπε

= = =

=

∫ ∫rr r

Coaxial cable

Capacitors in parallel: ...321 +++= CCCCtot

dA

VQC 0ε=Δ

=

Why it is obvious: capacitors in series and in parallel

1 20 0

...Q A A ACV d d

ε ε + += = ⇒Δ

Capacitors in series: ...1111

321

+++=CCCCtot

dA

VQC 0ε=Δ

=

Why it is obvious: capacitors in series and in parallel

0 01 2 ...

Q A ACV d d d

ε ε= = ⇒Δ + +

...1111

321

+++=CCCCtot

CQV

VQC =ΔΔ

=

Capacitors in series

VQCΔ

=

Capacitors in parallel ...321 +++= CCCCtot

Network of Capacitors

Capacitor as an energy storage

22

0 02 2

//

12 2

2 2

1/

Q X

V Q C F kXQ CV X F k

qdW vdq dq dW fdx kxdxC

q QW dq W kxdx kXC C

Q CVW potential energy

C k

UC

= == =

= = = =

⇔

= = = =

= = = =

∫ ∫

Energy density of the electric field

2 2

2

2 220 0

02

2 2

2

12 2 2

Q CVUC

U CVu energy densityvolume Ad

A V Vu Ed Ad d

ε ε ε

= =

= = =

= = =

Energy of the capacitor is, in fact, stored between the plates. The “vacuum” is filled with the electric field;electric field acts like elastic media.

Energy density of the electric field; spherical capacitor

22

0 0 20

1 1( ) ( )2 2 4

Qu r E rr

ε επε

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

2

2QW potential energy UC

= = =

Voltage, charge and energy

0 1 0 1 2

1 21 2

1 2

1 0 2 0

960

80

2 1640 3203 3

V C Q Q Q CQ QV V voltC C

Q Q C Q Q C

μ

μ μ

= = + ⇒

= = = =

= = = =

Moore’s Law (1965): every 2 years the number of transistors on a chip is doubled

Smaller, Denser, Cheaper

2 billion transistors per chip

Capacitance used in radio tuning is about hundred picofarads.

Capacitance used in cameras as a storage of energy for a flash light is of a few hundred microfarads.

2 2

2 2Q VU CC

= =

Why capacitance used in cameras is so large?

Why photo-cameras need a large capacitance ?Because a sources of energy is a battery with a given voltage V.

Most capacitors have a non-conducting material, or dielectric, between their conducting plates. When we insert an uncharged sheet of dielectric between the plates, the potential difference decreases to a smaller value V.

dielectricwithoutCQV

00 =

dielectricwithCQV =

0CC >

When the space between plates is completely filled by the dielectric, the ratio

0CCK = is called dielectric constant.

When the space between plates is completely filled by the dielectric, the ratio

0CCK = is called dielectric constant. K>1

Dielectric constant K

permittivity 0Kε ε=

K is called dielectric constant, K>1

Dielectric constant K

permittivity 0Kε ε=

to avoid confusion, don’t use V0

E σε

=

0CCK = is called dielectric constant. K>1

Dielectric constant K

0

212

Q A AC KV d d

E u E

ε ε

σ εε

= = =

= =

Part of the energy is stored by the polarizing molecules. Working with the same battery (fixed voltage source) one may accumulate larger charge. As a result, larger energy accumulation. Charging of the capacitor takes time.

permittivity 0Kε ε=

Additional MaterialsTransistors, Moore’s Law

Moore’s Law (1965): every 2 years the number of transistors on a chip is doubled

Smaller, Denser, Cheaper

2 billion transistors per chip

Si-MOSFET (metal-oxide-surface field-effect transistor)

gate voltage opens or closes the field effect transistor, which is recharging like any other capacitor

MOSFET: the workhorse of Integrated Circuits

Voltage, charge and energy

0 1 0 1 2

1 21 2

1 2

1 0 2 0

960

80

2 1640 3203 3

V C Q Q Q CQ QV V voltC C

Q Q C Q Q C

μ

μ μ

= = + ⇒

= = = =

= = = =

Intel switches to hafnium-based transistors for new 45-nm Penryn processors