Capacitance

The Ability to Store Charge

Common Types

Electrolytic Tanalum Silver Mica

Parallel Plate Capacitor

+

+++

++

――――――

Dielectric

d

Plate Area = A

Capacitance Depends on… Separation of the plates (d) [d in m]

d

1C

AC

C

Plate Area (A) [A in m2]

Dielectric Constant () [ in C2/N-m2]

Capacitance

d

AC

farad J

C

Nm

C

Nm

mC

m

mNmC

:Units

22

3

22

22

2

Capacitance depends on the construction of the capacitor, not on the circuit it is used in.

Alternate Form for C

V

C

C/J

C

J

CeCapacitanc

units the from2

Thus,

C = Q/V

Recall, V=W/Q

Example

What is the capacitance of a capacitor if the charge is 0.075 C held at V = 400 V

C = Q/V =0.075 C/400 V = 1.875 X 10-4 f = 187.5 f

Storage of Charge Charges flow from the wire to one plate of

the capacitor. The insulator (dielectric) prevents charges from flowing to the other plate. Charge accumulates.

The accumulated charge attracts opposite charges on the other plate.

The second plate charge pushes opposite charges down the other wire.

+

+++

++

――――――

Capacitors in DC Circuits

Capacitor in a dc circuit Capacitor will begin charging and current

will flow When fully charged (Q= CV), current

stops Electric field between plates cancels the

field produced by the battery voltage

Dielectric Constant

Many texts give in terms of a fundamental constant (o) and a number that depends on the material (K)

= Ko

o = permittivity of free space

= 8.854 X 10-12 C2/Nm2

K = dielectric constant

Typical Values of KMaterial K

Air (1 atm) 1.00059

Vacuum 1.00000

Ammonia 22

Glass 5-10

Mica 3-6

Paraffin Wax 2.1-2.5

Porcelain 6.0-8.0

Rubber 2.5-3.0

Example Two rectangular sheets of copper foil 16 X 20 cm are

separated by a thin layer of paraffin wax 0.2 mm thick. Calculate the capacitance if the dielectric constant for the wax is 2.4.

pf3400f10X4.3C

m10X2

)m032.0)(4.2)(Nm/C10X854.8(C

m10X2mm2.0d

m032.0cm320)cm20)(cm16(A

d

AKC

9

4

22212

4

22

o

Combining Capacitance

Parallel – plates of the capacitors are at the same voltage as the poles of the battery.

Voltage across AA’ = BB’ = E

Parallel Capacitance

Q1 = C1E Q2 = C2E Q3 = C3E

Total Charge stored is…

QT = Q1 + Q2 + Q3

QT = E(C1 + C2 + C3) = CTE

CT = C1 + C2 + C3

Series Capacitance

The same current flows through each capacitor

Current is charge/time, so Q1 = Q2 = Q3 = QT

Series Capacitance Kirchhoff’s Voltage Law

E = V1 + V2 + V3

321T

3

T

2

T

1

T

T

T

3

3

2

2

1

1

T

T

C

1

C

1

C

1

C

1

C

Q

C

Q

C

Q

C

Q

C

Q

C

Q

C

Q

C

Q

Try it

Adding Capacitance

Example

Three capacitors of 2.0, 3.0, and 5.0 f are connected in parallel to a 12 V source.1.Find the charge on each capacitor.2.Find the total charge of the combination.3.Find the total charge if the same three are connected in series to the 12 V source.

Solution (Parallel connection)

1. Q1 = VC1 = (12 V)(2.0X 10-6f) =24 C

Q2 = VC2 = (12 V)(3.0X 10-6f) =36 C

Q3 = VC3 = (12 V)(5.0X 10-6f) =60 C

2. QT = Q1 + Q2 + Q3 = (24+36+60)C

= 120 C

Solution (part 3) Series connection

f31

30C

30

31

30

61015

f0.5

1

f0.3

1

f0.2

1

C

1

C

1

C

1

C

1

T

321T

C6.11Q

)f10X31

30)(V12(VCQ

T

6TT

RC Circuits

We can apply Kirchhoff’s Laws to this circuit also

E = VC + VR = Q/C + IR

Charge Form For a dc voltage flowing for a time t

I = Q/t E = Q/C + QR/t = Q(1/C + R/t)

RCt

ECtQ

CtRCt

E

tR

C

EQ

1

Current Form Q = It E = Q/C + IR = It/C + IR = I(t/C + R)

RCt

ECI

CRCt

EI

Time dependent forms Notice that RC must have units of time –

most books call this = RC Current

/teR

Ei

)e1(EV /t Voltage

Saving Links

Charging Circuits RC Circuit Applet