Capacitors

• A capacitor is a device that stores electric charge.

• A capacitor consists of two conductors separated by an insulator.

• Capacitors have many applications:– Computer RAM memory and keyboards.– Electronic flashes for cameras.– Electric power surge protectors.– Radios and electronic circuits.

Types of Capacitors

Parallel-Plate Capacitor Cylindrical Capacitor

A cylindrical capacitor is a parallel-plate capacitor that hasbeen rolled up with an insulating layer between the plates.

Capacitors and Capacitance

Charge Q stored:

CVQ The stored charge Q is proportional to the potential

difference V between the plates. The capacitance C is the constant of proportionality, measured in Farads.

Farad = Coulomb / Volt

A capacitor in a simpleelectric circuit.

Parallel-Plate Capacitor

• A simple parallel-plate capacitor consists of two conducting plates of area A separated by a distance d.

• Charge +Q is placed on one plate and –Q on the other plate.

• An electric field E is created between the plates.

+Q -Q

+Q -Q

Energy Storage in Capacitors• Since capacitors store electric charge, they store

electric potential energy.• Consider a capacitor with capacitance C, potential

difference V and charge q.• The work dW required to transfer an elemental

charge dq to the capacitor:

dqc

qVdqdW The work required to charge the

capacitor from q=0 to q=Q:

222

0

2

00 2

1

2

)(

22

1CV

C

CV

C

Qq

Cdq

C

qVdqW

QQQ

Energy Stored by a Capacitor = ½CV2 = ½QV

Example: Electronic Flash for a Camera

• A digital camera charges a 100 μF capacitor to 250 V.

a) How much electrical energy is stored in the capacitor?

b) If the stored charge is delivered to a krypton flash bulb in 10 milliseconds, what is the power output of the flash bulb?

Dielectrics• A dielectric is an insulating material (e.g.

paper, plastic, glass).

• A dielectric placed between the conductors of a capacitor increases its capacitance by a factor κ, called the dielectric constant.

C= κCo (Co=capacitance without

dielectric)

• For a parallel-plate capacitor:d

A

d

AC 0

ε = κεo = permittivity of the material.

Properties of Dielectric Materials• Dielectric strength is the maximum electric field that a

dielectric can withstand without becoming a conductor.• Dielectric materials

– increase capacitance.– increase electric breakdown potential of capacitors.– provide mechanical support.

MaterialDielectric Constant κ

Dielectric Strength (V/m)

air 1.0006 3 x 106

paper 3.7 15 x 106

mica 7 150 x 106

strontium titanate 300 8 x 106

Practice Quiz

• A charge Q is initially placed on a parallel-plate capacitor with an air gap between the electrodes, then the capacitor is electrically isolated.

• A sheet of paper is then inserted between the capacitor plates.

• What happens to:a) the capacitance?b) the charge on the capacitor?c) the potential difference between the plates?d) the energy stored in the capacitor?

Discharging a Capacitor

• Initially, the rate of discharge is high because the potential difference across the plates is large.

• As the potential difference falls, so too does the current flowing

• Think pressure

As water level falls, rate of flow decreases

• At some time t, with charge Q on the capacitor, the current that flows in an interval t is:

I = Q/t

• And I = V/R• But since V=Q/C, we can say that

I = Q/RC• So the discharge current is proportional to

the charge still on the plates.

• For a changing current, the drop in charge, Q is given by:

Q = -It (minus because charge Iarge at t = 0 and falls as t increases)

• So Q = -Qt/RC (because I = Q/RC)

• Or -Q/Q = t/RC

• If we make the change infinitesimally small we get…

Q

Q

dQQ

dtRC

dQQ

dtRC

Q

Q

t

Q

Q

t

0

0

0

lnRC

t

becomes This

11

as same theisWhich

11

0

0

• Applying the laws of logs

0

0

0

Qln intecept and 1/RCgradient of linestraight a give willagainst t lnQ ofgraph A

c mx y

linestraight a ofequation

with the thisCompare

lnlnRC

t

Becomes

ln

QQ

Q

Q

RC

t

• Of course, we can’t easily measure Q but since Q = VC, and C is just constant, we can measure V and plot a graph of ln V against t.

ln V

t

Gradient = 1/RCln V0

• Alternatively, if you want to get rid of the ln, just multiply both sides by e.

ln Q = ln Q0 –t/RC

Becomes

Q = Q0e-t/RC

And gives a graph that looks like this…

Incidentally, graphs can be V or Q against t, they all have the same basic shape

The time constant.

• The time constant =RC.

• The units are seconds (t/RC is dimensionless).

• The time taken for the charge to fall to (1/e) of the initial value in the circuit.

• The time taken for the voltage to fall by (1/e) of its initial value.

Charging a capacitor

• At time t=0 the switch is closed, with the capacitor initially uncharged.

• A current will flow =Vc+VR=I0R, as initially Vc=0.

Thus the initial current is I0=/R.

• Now a charge begins to build on the capacitor, introducing a reverse voltage. The current falls,

and stops when the P.D. across C is .

• Final charge is given by "Q=CV" => Q0=C.

Charging a capacitor (quantitative).

• Apply Kirchoff's loop rule.

dtdQ

CQ

CQ

Rc RiRVV QCRCC

QRdt

dQ 11

RCdt

QCdQ

ttRC

QQ tQC 0

10)ln(

RCtCQC )ln()ln(

Charging a capacitor (cont)

• Where Q0 = C = the final charge on the capacitor.

RC

t

C

QC

ln

RCteC

QC /

RCteQ

Q /

0

1

)1( /0

RCteQQ

Charging a capacitor (cont).

• To find the current, differentiate since I=dQ/dt.

• By considering time zero, when the current is I0,

RCtRCt eR

eRC

QI //0

./0

RCteII

)1( /0

RCteQQ

Energy Considerations.

• During charging, a total charge Q=C flows through the battery.

• The battery does work W=Q0=C2.

• The energy stored in the capacitor is ½QV=½Q0 =½C2.

• Where's the other half?

Finally…• Electromagnetism depends a lot on

integrals, vectors etc. shows how useful they are.

• It is one of the foundations of physics but:– it can be rather formal, encouraging the

precise thinking that we expect of any academic training;

– it is rather far removed from the everyday, but that develops the imagination we expect from a physicist.