Capacitors and capacitance - Texas A&M...
Transcript of Capacitors and capacitance - Texas A&M...
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