Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the...

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Lesson 6 Capacitors and Capacitance
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Transcript of Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the...

Page 1: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Lesson 6Capacitors and

Capacitance

Page 2: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 3: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 1Capacitance, Charge, and

Voltage

Page 4: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 5: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

+

Page 6: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

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

Page 7: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

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

Page 8: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 9: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 10: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Charging a CapacitorWe find that voltage is proportional to charge.

Q

V

Page 11: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Charging a CapacitorWe find that voltage is proportional to charge.

Q

V

CV

Qslope

Page 12: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Capacitance

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

Q=CV

Page 13: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 2Parallel Plate Capacitors

Page 14: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 15: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• The electric field is the sum of the electric fields of a positively charged palate …

Page 16: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• … and a negatively charged plate.

Page 17: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• … and a negatively charged plate.

Page 18: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• The electric fields outside the plates cancel out.

Page 19: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

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

Page 20: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• The electric fields between the plates add.Just make the arrows align…

Page 21: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

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

Page 22: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

• The electric field inside is uniform.•The electric field outside is small.

Page 23: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Plate Capacitor

Page 24: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Electric Field of a Capacitor

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

Page 25: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Electric Field of a Capacitor

The field lines inside a capacitor:

Page 26: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

The field lines inside a capacitor:

Electric Field of a Capacitor

Page 27: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

point charge with a charge Q.

capacitor with a charge Q and plate area A

+ + + + + + + +

Electric Field of a Capacitor

Page 28: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

+ + + + + + + +

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.

Page 29: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

A

QE

A

NkE

0

0

20

2

4

1)(

4)(,

QkN

r

QrE

r

NkrE

A

NkE

←same N →

N lines between the plates!

+ + + + + + + +

Electric Field of a Capacitor

r

Page 30: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

A

QE

0

+ + + + + + + +

Electric Field of a Capacitor

Page 31: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

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.

A

QdEdV

0

Page 32: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

d

A

V

QC 0

d

Now we can find the capacitance:

Page 33: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate Capacitors

d

A

V

QC 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.

Page 34: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Parallel-plate CapacitorEquations

d

AC

EdV

CVQ

0

Page 35: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 4Dielectrics

Page 36: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Dielectrics

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

Page 37: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Dielectrics

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

Page 38: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Dielectrics

•The net charge inside the dielectric is zero.

Page 39: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Dielectrics

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

Page 40: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Dielectrics

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

E of dielectric

E of plates

Page 41: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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 )(

CV

Q

V

QC

QCVQ

VdEEV d

Page 42: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

With the dielectric:

00

0

0

00 )(

CV

Q

V

QC

QCVQ

VdEEV d

0

0

0

CC

VV

QQ

Page 43: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

0

0

0

CC

VV

QQ

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

1

Page 44: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

VCCVQ

VV

Page 45: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Problem Type 2:Fixed Voltage

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

With the dielectric:

00

0

VCCVQ

VV

0

0

0

CC

VV

QQ

Page 46: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Problem Type 2:Fixed Voltage

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

0

0

0

CC

VV

QQ

The charge on the

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

Page 47: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 5Energy in Capacitors and

Electric Fields

Page 48: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 49: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 50: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

dqCd

qEdqF )(

Page 51: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Energy in a Capacitor

•The work done in moving the charge is

qdqC

FddW1

Page 52: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Energy in a Capacitor

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

UCVC

Qqdq

CdWW

Q

22

0 2

1

2

1

Page 53: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Energy in a Capacitor

2

2

1CVU

Page 54: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Energy Density

Energy per unit volume in a an electric field.

In a parallel-plate capacitor of volume v=Ad : AdEEd

d

ACVU 2

0202

2

1

2

1

2

1

202

1E

v

Uu

Page 55: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Energy Density

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

202

1Eu

Page 56: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 57: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 6Capacitors in DC Circuits

Page 58: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Capacitors in Circuits

•In DC circuits, capacitors just charge or discharge.

•No current flows after a capacitor is fully charged or discharged.

Page 59: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

20 μF

12 V

5 Ω

1 Ω

2 Ω

Page 60: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 61: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 62: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 63: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 64: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 65: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 66: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 7Capacitors in Series and

Parallel

Page 67: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Adding Capacitors

Resistors:

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!

Page 68: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Adding Capacitors

Series:

Parallel:

212121

111,,

CCCVVVQQQ

212121 ,, CCCQQQVVV

Page 69: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.)

Page 70: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Why are the charges the same on capacitors in series?

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

Page 71: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Why are the charges the same on capacitors in series?

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

Page 72: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Why are the charges the same on capacitors in series?

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

I

q qq q

Page 73: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Why are the charges the same on capacitors in series?

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

+Q –Q

Page 74: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Why are the charges the same on capacitors in series?

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

Page 75: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

Section 8Charging and Discharging Capacitors and the Time

Constant

Page 76: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 77: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

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

0 IRC

Q

I

Vol

tage

C

QVC

IRVR

I

Page 78: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

The minus sign comes from:

1) I > 0

2) Q is the charge on the capacitor

3) The capacitor is discharging so

dt

dQI

QI

IRC

Q

0,0

0

0dt

dQ

I

Vol

tage

C

QVC

IRVR

Page 79: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

)(1

0

0,0

0

tQRCdt

dQdt

dQR

C

Qdt

dQI

QI

IRC

Q

I

Vol

tage

C

QVC

IRVR

Page 80: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

)(1

tQRCdt

dQ

•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

Q

Qdt

RCQ

dQdt

RCQ

dQln

11

Page 81: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

/0

0

0

)(

)(

,1)(

ln

1

1

0

t

ttQ

Q

eQtQ

RCtRCQ

tQ

dtRCQ

dQ

dtRCQ

dQ

Page 82: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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.

Page 83: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Discharging

τ=1 sτ=2 s

τ=3 se

1

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.

Page 84: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

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

Page 85: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Charging

•We again use Kirchoff’s Loop rule:

00 C

QIRV

I

Vol

tage

C

QVC

IRVR

0V

Page 86: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Charging

•We again use Kirchoff’s Loop rule:

0

0

0

0

0

C

Q

dt

dQRV

dt

dQI

C

QIRV

I

Vol

tage

C

QVC

IRVR

0V

Page 87: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Charging

•This differential equation has the solution:

00 C

Q

dt

dQRV

RC

CVQ

eQtQ

f

tf

0

/1)(

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

Page 88: Lesson 6 Capacitors and Capacitance. Class 16 Today, we will: learn what a capacitor is. learn the definition of capacitance. find the electric field.

RC Charging

τ=1 s

τ=2 s

τ=3 s

e

11

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.