Capacitors, Multimeters, Circuit Analysis

Capacitor: Charging and Discharging

Charging Discharging

The capacitor in your set is similar to a large two-disk capacitor

s

D

Capacitor: Construction and Symbols

There is no connecting path through a capacitor

Positive and negative charges are attracted to each other: how can they leave the plates?

Fringe field is not zero!

How is Discharging Possible?

Electrons in the wire near the negative plate feel a force thatmoves them away from the negative plate.

   

Electrons near the positive plate are attracted towards it.

Capacitor: Charging

Ultimately, the fringe field of the capacitor and the fielddue to charges on the wire are such that E=0 inside the wire. At this point, i=0.

Why does current ultimately stop flowing in the circuit?

Thick filamentThin filament

Which light bulb will glow longer?

Why?

1) Round is brighter capacitor gets charged more?

2) Long bulb glows longer capacitor gets charged more?

The Effect of Different Light Bulbs

Will it glow at all?How do electrons flow throughthe bulb?

An Isolated Light Bulb

Why do we show charges nearbulb as - on the left and + onthe right?

Electric field in a capacitor:0

/

AQ

E

E

s

+Q-Q

f

i

ldEV

EsV

sAQ

V0

/

V

s

AQ 0

VQ ~In general:

Definition of capacitance:VCQ

Capacitance

Capacitance of a parallel-plate capacitor:

s

AC 0

Capacitance

Capacitance

Michael Faraday

(1791 - 1867)

VCQ

Units: C/V, Farads (F)

The capacitor in your set is equivalent to a large two-disk capacitor

s=1 mm

D

How large would it be?

s

AC 0

0

CsA

A 1 F 0.001 m

9 10 12 C2 /N m2 A 1.1108 m2

D ~ 10 km (6 miles)

Exercise

A capacitor is formed by two rectangular plates 50 cm by 30 cm, and the gap between the plates is 0.25 mm. What is its capacitance?

s

AC 0

C 0A

s

9 10 12 C2 /N m2 0.15 m2 2.5 10 4 m

5.4 10 9 F 5.4 nF

Exercise

s

D

No insulator:

0

/

AQ

E

EsV

sAQ

V0

/

Vs

AQ 0

s

AC 0

With insulator:

0

/

K

AQE

EsV

sK

AQV

0

/

Vs

AKQ 0

s

AKC 0

A Capacitor With an Insulator Between the Plates

Ammeter: measures current I

Voltmeter: measures voltage difference V

Ohmmeter: measures resistance R

Ammeters, Voltmeters and Ohmmeters

0.150

Connecting ammeter:

Conventional current must flow into the ‘+’ terminal and emerge from the ‘-’ terminal to result in positive reading.

Using an Ammeter

Simple ammeter using your lab kit:

Simple commercialammeter

Digital ammeters: uses semiconductor elements.ADC – analog-to-digital converter(Combination of comparator and DAC)

Ammeter Design

VAB – add a series resistor to ammeter

R

VI

Measure I and convert to VAB=IR

Connecting Voltmeter:

Higher potential must be connected to the ‘+’ socket and lower one to the ‘-’ socket to result in positive reading.

VoltmeterVoltmeters measure potential difference

R

How would you measure R?

A

Ohmmeter

R

emfI

I

emf

Ohmmeter

Ammeter with a small voltage source

0_ Ctripround VRIemfV

C

QVC

0C

QRIemf

I dQ

dt

emf Q / C

R

Initial situation: Q=0R

emfI 0

Q and I are changing in timedt

d

RC

Q

dt

d

R

emf

dt

d

dt

dIdt

dQ

RCdt

dI 1 I

RCdt

dI 1

Quantitative Analysis of an RC Circuit

IRCdt

dI 1

dtRC

dII

11

tI

I

dtRC

dII 0

11

0

RC

tII 0lnln

RC

t

I

I

0

ln

RC

t

eI

I

0

RCteII /0

Current in an RC circuit

What is I0 ?

RCteR

emfI /

Current in an RC circuit

RC Circuit: Current

What about charge Q?

RCteR

emfI /

Current in an RC circuit

dt

dQI

IdtdQ

t

RCtt

dteR

emfIdtQ

0

/

0

Q C emf 1 e t / RC

RC Circuit: Charge and Voltage

 

   

RCteR

emfI /

Current in an RC circuit

Charge in an RC circuit

Q C emf 1 e t / RC

Voltage in an RC circuit

V emf 1 e t / RC

RC Circuit: Summary

RCteR

emfI /

Current in an RC circuit

When time t = RC, the current I drops by a factor of e.

RC is the ‘time constant’ of an RC circuit.

37.0718.2

11/ ee RCt

The RC Time Constant

A rough measurement of how long it takes to reach final equilibrium

What is the value of RC?

About 9 seconds

Question

A 0.5 farad capacitor isconnected to a 1.5 voltbattery and a bulb, andcurrent runs until the bulbgoes out. What is theabsolute value of thecharge on one plate of thecapacitor?

A) 0.33 CB) 0.5 CC) 0.75 CD) 1.5 CE) 3.0 C

A B

C D

What is the final charge on a 1 F capacitor connected to a 1.5V battery through resistor 100 ?

C 5.1 VCQ

Can you apply the RC equations to the circuit below?

RCteR

emfI /

Current in an RC circuit

No! Resistance depends on current.

Exercise

Find currents through resistors

Loop 1

Loop 2

Loop 3

Loop 4

I1

I2

I3

I5I4

loop 1:017441111 IRIRIRIremf

loop 2:

033262222 IRIRemfIrIR

loop 3:

0553344 IRIRIR

nodes:04321 IIII

0523 III

0154 III

Five independent equations and five unknowns

Exercise: A Complicated Resistive Circuit