Lesson 6 Capacitors and Capacitance

Lesson 6Capacitors and

Capacitance

Class 16Today, we will:• learn what a capacitor is.• learn the definition of capacitance.• find the electric field and voltage inside a parallel-plate capacitor.• find the capacitance of the capacitor.• learn that a dielectric is a material with polar molecules.• learn how dielectrics increase capacitance.• find the energy stored in a capacitor and in the electric field.

Section 1Capacitance, Charge, and

Voltage

What is a Capacitor?•Conductors that can hold charge.•Cables, hands, etc. all have capacitance.•For our purposes: two conductors, one with charge +Q and one with charge −Q.

What is a Capacitor?•We “charge” a capacitor by connecting it to a battery.

+


•When we disconnect the battery, charge remains on the conductors.


•When we disconnect the battery, charge remains on the conductors.•If we connect the conductors, charge will then flow from one to the other.

Why Are Capacitors Useful?

•Capacitors can provide uniform electric fields. We use them to accelerate or deflect charged beams, etc.•We can store charge for later use.•We can charge many capacitors and then discharge them at one time to produce very large currents for a short time.•Capacitors are important in AC (alternating current = sinusoidal) circuits, but we’ll study that later.

Charging a CapacitorWhen we attach a capacitor to a battery:•Charge builds up on the conductors.•The charge on the + conductor is equal and opposite the charge on the − conductor.•We call +Q the “charge on the capacitor” •Voltage builds up on the capacitor until it has the same voltage as the battery. •Electric field builds up in the capacitor.

Charging a CapacitorWe find that voltage is proportional to charge.

Q

V

Charging a CapacitorWe find that voltage is proportional to charge.

Q

V

CVQslope

Capacitance

•If capacitance is large - - the capacitor holds a large charge at a small voltage.

Q=CV

Section 2Parallel Plate Capacitors

Parallel-plate Capacitors• made of two plates each of area A (the shape doesn’t matter) •plates are separated by a distance d.

Parallel-plate Capacitors• The electric field is the sum of the electric fields of a positively charged palate …

Parallel-plate Capacitors• … and a negatively charged plate.

Parallel-plate Capacitors• … and a negatively charged plate.

Parallel-plate Capacitors• The electric fields outside the plates cancel out.

Parallel-plate Capacitors

• The electric fields outside the plates cancel out.Make the outside fields disappear.

Parallel-plate Capacitors• The electric fields between the plates add.Just make the arrows align…

Parallel-plate Capacitors• The charges move to the inside of the plates.Move the + and – symbols toward the center.

Parallel-plate Capacitors• The electric field inside is uniform.•The electric field outside is small.

Section 3Electric Field, Voltage, and Capacitance in a Parallel-

Plate Capacitor

Electric Field of a CapacitorWe can find the electric field in a capacitor from Coulomb’s law and our knowledge of field lines!

Electric Field of a CapacitorThe field lines inside a capacitor:

The field lines inside a capacitor:

Electric Field of a Capacitor

point charge with a charge Q.

capacitor with a charge Q and plate area A

+ + + + + + + +


+ + + + + + + +

Electric Field of a Capacitor•Field lines begin on the positive charge in both cases.

•Since the positive charge is the same, the number of field lines is the same.

AQE

ANkE

0

0

20

2

41)(

4)(,

QkN

rQrE

rNkrE

ANkE

←same N →

N lines between the plates!

+ + + + + + + +


r

AQE0

+ + + + + + + +



EdEdxV

We always ignore the minus sign, so V will be positive:

d

We know how the voltage relates to the electric field because the electric field is constant.

AQdEdV

0


dA

VQC 0

d

Now we can find the capacitance:


dA

VQC 0

d

Now we can find the capacitance:

•If the plate area is large, the capacitor can hold more charge.•If the plate separation is small, the charges on the two plates attract each other with a stronger force, so thecapacitor can hold more charge.

Parallel-plate CapacitorEquations

dAC

EdVCVQ

0

Section 4Dielectrics

Dielectrics•A dielectric is an insulator with polar molecules that is placed between the plates of a capacitor.

Dielectrics•Polar molecules rotate in the electric field of the capacitor.

Dielectrics•The net charge inside the dielectric is zero.

Dielectrics•But there is leftover charge on the surfaces of the dielectric.

Dielectrics•This charge produces an electric field that opposes the electric field of the plates.

E of dielectric

E of plates

Problem Type 1:Fixed Charge

•A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

Without the dielectric: With the dielectric:

000 VCQ

00

0

0

00 )(

CVQ

VQC

QCVQ

VdEEV d



With the dielectric:

00

0

0

00 )(

CVQ

VQC

QCVQ

VdEEV d

0

0

0

CC

VV

QQ



0

0

0

CC

VV

QQ

The electric field of the dielectric reduces the voltage across the capacitor, causing the capacitance to rise.

1

Problem Type 2:Fixed Voltage

•A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.

Without the dielectric: With the dielectric:

00VCQ

00

0

VCCVQVV



With the dielectric:

00

0

VCCVQVV

0

0

0

CCVV

QQ



0

0

0

CCVV

QQ

The charge on the

dielectric pulls additional charge from the battery to the plates, causing the capacitance to rise.

Section 5Energy in Capacitors and

Electric Fields

Energy in a Capacitor•Start with two parallel plates with no charge.•Move one charge from one plate to the other.•There is no electric field and no force, so it requires no work.

Energy in a Capacitor•After the charge is transferred, the capacitor has a small charge and a small field.•The field causes a force on the next charge we move, forcing us to do work.

Energy in a Capacitor•When the charge on a capacitor is q, the voltage is q/C and the electric field is V/d=q/Cd.•The force on a small charge dq is

dqCdqEdqF )(

Energy in a Capacitor•The work done in moving the charge is

qdqC

FddW 1

Energy in a Capacitor•The work done in charging the capacitor to its final charge Q is:

UCVC

QqdqC

dWWQ

22

0 21

21

Energy in a Capacitor

2

21 CVU

Energy DensityEnergy per unit volume in a an electric field.In a parallel-plate capacitor of volume v=Ad : AdEEd

dACVU 2

0202

21

21

21

202

1 EvUu

Energy DensityThe density of the energy stored in any electric field, not just a capacitor, is:

202

1 Eu

Class 17Today, we will:• learn how to combine capacitors in series and parallel• find that circuits RC circuits have charges and currents that depend on exponential functions• learn the meaning of the exponential time constant• find that the exponential time constant for an RC circuit is τ=RC

Section 6Capacitors in DC Circuits

Capacitors in Circuits•In DC circuits, capacitors just charge or discharge. •No current flows after a capacitor is fully charged or discharged.

Capacitors in Circuits•Describe what happens in this circuit after the switch is closed.

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits•Initially positive charge on the right plate of the capacitor pushes charge off the left plate. It is as if the capacitor were replaced by a wire.

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits•When the capacitor starts charging, it behaves like a battery that opposes the flow of current.

+─

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits•Eventually, the capacitor becomes fully charged. No more current flows. What is the final voltage on the capacitor?

+─

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits•First, ignore the branch with the capacitor. •Rtotal=3 Ω. I = 4 A. V across the 1 Ω resistor is IR = 4 V.

+─

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits•V across the 5 Ω resistor is 0. Why?•V across the capacitor is 4 V. •Q on the capacitor CV = 80 μC

+─

20 μF

12 V

5 Ω

1 Ω

2 Ω

Capacitors in Circuits

•Summary:

In steady state, no current flows through the capacitor. Just find the voltage across the capacitor and you can determine the charge.

Section 7Capacitors in Series and

Parallel

Adding CapacitorsResistors:

Capacitors:

IRV

CQV 1

CR 1

R and 1/C enter the voltage equations in a similar way.If you replace R with 1/C in series-parallel equations for resistors, you get the correct result for capacitors!

Adding CapacitorsSeries:

Parallel:

212121

111,,CCC

VVVQQQ

212121 ,, CCCQQQVVV

Let’s look at the voltage and charge equations…

•As with resistors, the voltages across two capacitors in series add to get the total voltage.

•As with resistors, the voltages across two capacitors in parallel are the same.

•When we discharge two capacitors in parallel, the total charge that leaves the capacitors is the sum of the charges. (Recall that with resistors the sum of the currents is the total current in parallel.)

Why are the charges the same on capacitors in series?

To begin with, there is no charge on either capacitor.


Before we start charging the two capacitors, the charge within the dashed box is zero.


A the capacitors charge, the charge within the dashed box remains zero.

I

q qq q


When the left plate of the left capacitor acquires its final charge +Q, the right plate’s charge is –Q.

+Q –Q


The charge within the box must remain zero, so the right capacitor must have the same charge as the left capacitor.

+Q –Q +Q –Q

Section 8Charging and Discharging Capacitors and the Time

Constant

RC Discharging•Charge a capacitor with a battery to a voltage V.

•Disconnect the capacitor and attach it to a resistor.

•The initial charge is Q=CV.

•The charge decays to zero – but what is Q(t)?

Q(t)

Q0

I

t

RC Discharging•Look at the voltage around the circuit. We can use Kirchoff’s loop rule:

0 IRCQ

I

Vol

tage

CQVC

IRVR

I

RC Discharging

The minus sign comes from:

1) I > 0

2) Q is the charge on the capacitor

3) The capacitor is discharging so

dtdQI

QI

IRCQ

0,0

0

0dtdQ

I

Vol

tage

CQVC

IRVR

RC Discharging

)(1

0

0,0

0

tQRCdt

dQdtdQR

CQ

dtdQI

QI

IRCQ

I

Vol

tage

CQVC

IRVR

RC Discharging)(1 tQ

RCdtdQ

•This is a differential equation, but it is a really easy one to solve.

•Usually we’ll just give you the solutions to differential equations.

i

f

QQ

dtRCQ

dQdtRCQ

dQ ln11

RC Discharging

/0

0

0

)(

)(

,1)(ln

1

1

0

t

ttQ

Q

eQtQ

RCtRCQ

tQ

dtRCQ

dQ

dtRCQ

dQ

RC Time Constant•τ (tau) is called the “RC time constant.” τ = RC.

• τ has units of seconds.

•When τ is big, capacitors charge and discharge slowly.

•If R is large, not much current flows, so τ is big.

•If C is large, there is a lot of charge that has to flow, so τ is big.

RC Discharging

τ=1 sτ=2 s

τ=3 se1

Discharging capacitors with three different time constants.

The time constant is the time it takes the charge to drop to 1/e of its original value.

RC Charging•A capacitor is initially uncharged.

•Use a battery with voltage V0 to charge the capacitor.

The voltage increases to V0.

•The charge increases to Q=CV0.

V0

C

R

RC Charging•We again use Kirchoff’s Loop rule:

00 CQIRV

I

Vol

tage

CQVC

IRVR

0V

RC Charging•We again use Kirchoff’s Loop rule:

0

0

0

0

0

CQ

dtdQRV

dtdQI

CQIRV

I

Vol

tage

CQVC

IRVR

0V

RC Charging

•This differential equation has the solution:

00 CQ

dtdQRV

RC

CVQ

eQtQ

f

tf

0

/1)(

•Try plugging the solution into the differential equation and see if it works!

RC Charging

τ=1 s

τ=2 s

τ=3 s

e11

Charging capacitors with three different time constants.

The time constant is the time it takes the charge to rise to 1-1/e of its final value.

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Lesson 6 Capacitors and Capacitance