1

PHYS113 Electricityand Electromagnetism

Semester 2; 2002

Set B Notes (WK-10 final) Professor B. J.

FraserThis TRACE image was, I believe, taken during the windowincluding the large emissions of mid-July 2000. It does notillustrate a point about Satellite Anomalies, but is a lovely andfantastic image – like something for inclusion in Walt Disney’s“Fantasia”. It looks as if it could be a cosmic conductor risingfrom the surface of the Sun, ready to conduct some Wagnerianpiece, perhaps the “Ride of the Valkyries”.

This TRACE image was, I believe, taken during the windowincluding the large emissions of mid-July 2000. It does notillustrate a point about Satellite Anomalies, but is a lovely andfantastic image – like something for inclusion in Walt Disney’s“Fantasia”. It looks as if it could be a cosmic conductor risingfrom the surface of the Sun, ready to conduct some Wagnerianpiece, perhaps the “Ride of the Valkyries”.

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Uses in Technology

Accelerators (1929)(Giancoli Section 44.2, p1115) Van der Graaf HV Accelerator Works because E-field inside

Gaussian sphere is zero 1m sphere 3 x 106 V Up to 20 MV producedPrecipitators (See Figure shown) Remove dust and particles from

coal combustion -ve wire @ 40 - 100 kV E-field particles to wall > 99% effective.Photocopiers (1940) (Giancoli Example 21.5, p555) Image on +ve photoconductive

drum Charge pattern -ve toner pattern Heat fixing +ve paper.

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5. What is Capacitance?What is a Capacitor?

Charge-carrying conductors are surrounded by an electric field.

Field can do work on other charges.

Capacitance describes the energy stored in the electric field between 2 equal but oppositely charged conductors.

Unit = Farad, F, = C/V

Large unit, - usually use F, nF, pF. A capacitor is a device comprising

a pair of conducting surface, plates, carrying charge with a p.d. and a fixed separation between them.

charge on either

p.d. between themCapacitance =

VQ

C

4

Parallel Plate Capacitor

From Gauss’ law, for each plate:

For 2 || plates:

But electric field, E = V/d,

AQ

EQ

EA00 2

2

AQ

E0

AQd

V0

QdA

QVQ

C 0

dA

C 0

+ ++

+++

+ + -

+

-- -

---

-

--

+Q-Q

Area, Ad

5

Cylindrical Capacitor

Inner conductor radius = a, Inner conductor radius = b, length = l.

From Gauss’ law,

Thus:

b

a

ab sdEVV

rk

E2

ab

krdr

kdrEVb

a

b

a

r ln22

ab

lkQQ

VQ

Cln

2

ab

k

lVQ

Cln2

Important for Important for practical practical

capacitors & capacitors & shielded cablesshielded cables

2a 2b

l

+Q+Q

-Q-Q

6

Capacitance of an Isolated Sphere

Potential of a sphere is:

Thus, the capacitance is:

Example: Capacitance of the Earth is:

7.1 x 10-4 F

rQ

V04

rVQ

C 04

7

Energy in Capacitors

The electric field contains energy Work to move

a charge dq is: For total charge, Q:

The work is stored as potential energy in the electric field:

Since electric field is: E = V/d

dqCq

VdqdW

CQ

dqqC

dqCq

dWWQQ

2

1 2

00

222

22 QVCVCQ

U

volume2222

20

20220

2 EAd

EdE

dACV

U

energy

volumeEnergy density =

2volume

20EU

u i.e. energy i.e. energy E E22

8

Alternative Energy Storage

Worked Example An alternative energy

proposal is for the storage of energy in electric fields of capacitors. Find the E field required to store 1J in a volume of 1 m3 in a vacuum.

3-2

0 m J 2E

u

1-5

12-

0

C N 10 x 4.75

10 x 85.81 2

2

u

E

I.e. Large !I.e. Large !

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Capacitors in Electric Circuits

In circuits, capacitors appear in parallel or series combinations.

Parallel Capacitors: Charge is stored on the plates of

both capacitors.Qtotal = Q1 + Q2

Since Q = CV & each capacitor has the same p.d. across it:

VQQ

VQ

C totaltotal

21

21 CCCeq

C1

C2

+

+

+

-

-

-

Q1

Q2

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Capacitors in Electric Circuits

Series Capacitors: The magnitude of the charge on

each plate is the same, Q. The potential difference is

summed across the capacitors:Vbattery = V1 + V2

where V1 = Q/C1 , V2 = Q/C2 , etc.

21 CQ

CQ

CQ

eq

21

111CCCeq

+-

C1 C2

+ -+ -

V1 V2

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Dielectrics

Dielectric - nonconducting material between the plates of a capacitor.

Examples: air, paper, plastic, glass. It has 2 important properties:

Dielectric Strength: Size of E-field (V/m) that causes

dielectric to fail (stop insulating). Arc or short circuit (Typ. ~ 106 V/m) Correct dielectric can increase max

operating voltage of capacitor.

Dielectric Constant: Molecular dipoles in the dielectric

material align with the electric field. Reduces effective field to E/,

where is a constant.

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Dielectric Constant

The capacitance therefore increases as well:

C = C0

where = dielectric constant This allows capacitors to be made

smaller by using high dielectrics. The energy density in the electric

field is also reduced to:

u = u0 /

Arises because it takes work to insert the dielectric

Piezo-electricity.

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6. Electric Currents

What is an electric current? An electric current is an organised

movement of charges. Usually but not always electrons. Charges move due to applied E-field. By definition,

average current:

and at any time, instantaneous current is:

Unit: 1 Ampere, A = C/s (‘amp’) Typ. household currents ~ few amps In electronic circuits, ~ mA, mA, nA. By convention, direction of current

flow chosen for +ve charges. I.e from +ve to -ve. Electrons actually moving other way.

tQ

Iav

dtdQ

I

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Currents in Materials(or, Why Do Lights Turn On?)

Current comprises charges flowing across an area dA at velocity v:

where J = current density = I/A for small A.

The number of charges passing through A is n x A.

And: J = n q vd

where vd = drift velocity of charges

and: n = number density.

dA

v

A

AdJI

AvnqtQ

I d

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Current in a Wire

Worked Example A light draws a current of 0.5 A

through a copper wire of diameter 1.0 mm. Find the drift velocity of electrons in the wire. The density of copper is 8.92 g cm-3.

What is n?

nqAI

vAvnqI dd

3-28

3-22

23

Cu

m electrons 10 x 8.5

cm electrons 10 x 8.5

63.58.9210 x 6.02

M

densityunit electrons ofnumber

CuAN

n

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Current in a Wire

A = cross-sectional area of wire

= r2

= (5 x 10-4) 2

Thus, it takes 6 hours for an electron to move 1 m.

Why do the lights turn on so quickly?

1-7-

7-19-28

2

ms 10 x 4.7

10 x 2.5 10 x 1.610 x 8.55.0

rnqI

nqAI

vd

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Resistance and Conductivity:Ohms ain’t Ohms!

The rate at which charges move in a conductor due to an electric field depends on magnitude of the field.

Thus: J = E where = conductivity &

depends on geometry & properties of conductor.

This is known as Ohm’s Law. Not all materials obey Ohm’s law. I.e. not all materials are linear. Metals at increasing temperature &

semiconductors don’t obey Ohms law.

These are non-ohmic conductors.

For a wire of length l & area A.E = V / lE = J / l

VAI

J

Avd

l

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Conductivity of Materials

where = resistivity = 1 / Thus: V = I R where R = l/A = resistance Unit = Volt / Amp: = V / A

A resistor is a device built with a specified resistance ( ‘s M’s)

Units of resistivity are .m, & conductivity ( .m)-1. (mho).

Good conductor has low .

IAl

IAl

V

1

Ohmic conductor(e.g. resistor)

Non-ohmic conductor(e.g. diode)

I I

VV

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Electric Poweror, How Bright is your Light?

Charges lose energy in flowing in a material (supplied by the battery).

For a small charge dq moving through a p.d. V,

dU = V dq Thus, power is given by:

But I = dq/dt: P = VI Unit = Watt, 1W = 1 J/s For an ohmic material the power

dissipated (mostly in heat) is:

dtdq

VDtdW

P

RV

RIIVP2

2

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Car Starter Motor

Worked Example A car starter motor draws 500 A

through a wire of resistance 0.01 . Find the voltage drop and the power loss in the cable.

V = IR = 500 x 0.01 = 5 V

P = I2 R = (500)2 x 0.01 = 2 500 W

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7. Direct Current CircuitsSources of EMF

The electric energy that drives charges around a circuit is called the electromotive force (emf)

Not a force but an energy.

A source of emf increases the potential energy of charges in a circuit (“pumps them up”)

By definition, the emf () is given by:

Unit is the Volt (= J/C)

dqdW

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Equivalent Circuits and Thevenin’s Theorem

All circuits, no matter how complex can be reduced to a simple equivalent circuit

This circuit has a source of emf, , and a resistance, R.

This is Thevenin’s theorem

The net potential energy around the circuit is:

- I R = 0

+-

I

R

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Internal Resistance

All real sources of emf have some internal resistance that:

Reduces the output terminal voltage

Limits the power that can be delivered by the emf source.

+-

I

R

Rint

V = - I Rint

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Sources of EMF

Name ConvertsBattery Chemical Electrical

Generator Mechanical Electrical

Solar Panel Radiation Electrical

Thermocouple Heat Electrical

MHD Magnetic Electrical

Potential around a circuit:

V

+-

r Rb

c

d

a

a'

a a' b c d

IrIR

I

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Resistors in Circuits

Combinations of resistances in circuits may be in series or parallel.

Resistors in series Same current in each resistor. Voltage across each is:

Vr = I R

Around the circuit loop.V = I (R1 + R2)

Therefore:

+-

R1 R2

V1 V2

V

II

I

Req = R1 + R2

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Resistors in Circuits

Resistors in parallel Same p.d. across each resistor. Current is shared between

resistors.

In household circuits appliances are connected in parallel.

Xmas tree lights are often in series.

2121

11RR

VIII

R1

R2

+-

I1

I2

VI 21

111RRReq

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Circuit Analysis: Kirchoff’s Laws

Complex circuits involving multiple loops are analysed using Kirchoff’s Laws.

First Law: At a JunctionAt a Junction The sum of the currents entering

and leaving the junction is zero. Statement of conservation of

charge

Second Law: Around a Circuit Around a Circuit LoopLoop

The sum of potential changes is zero

The potential is conserved. Statement of conservation of

energy

outin II

0loop

V

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Circuit Analysis Example

Worked Example Find the current in each branch of

the circuit shown below.

2 +

-

+-

3

1 4

10

10 V

5 V

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Circuit Analysis Solution

Pick a junction and assign arbitrary current directions and sum to zero.

It doesn’t matter if the initial guess of current direction is wrong since the answer will just be a -ve value!2

1

10

I1

I2I3

I1 + I2 = I3 (1)(1)

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Circuit Analysis Solution

Sum potential drops around first loop.

Mark all voltage rises & drops depending on the current.

Current flow from +ve to -ve !10 - 2 I1 + I2 - 5 + 4 I2 - 3 I1 = 0

2 +

-

+-

3

1 4

10 V

5 V

I2

I1

++

++

--

++-- --

++--a

- 5 I1 + 5 I2 - 5 = 0

I1 - I2 = 1 (2)(2)

Start here

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Circuit Analysis Solution

Sum around the other loop.

Solve simultaneously & check !

I1

I3

2 +

- 3 10 V

++-- ++--

10

++ --

10 - 2 I1 -10 I3 - 3 I1 = 0

- 5 I1 - 10 I3 + 10 = 0

I1 + 2 I3 = 2 (3)(3)

II1 1 = 0.8 A, I= 0.8 A, I22 = - 0.2 A, I = - 0.2 A, I33 = 0.6 A = 0.6 A

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Maximum Power Transfer

Practically we are interested in the amount of power that can be transferred from source to load.

The max. amount of power will be transferred from any source (with internal resistance, r) to a load (of resistance, R) when R equals r.

Recall that:

the power delivered to the load is:

When is P a maximum?

+-

I

R

Rint

rRI

R

rRRIP 2

22

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Maximum Power Transfer

Easiest to plot P as a function of R/r.

Can also calculate dP/dx = 0

where x = R/r but this is tricky!

Max value when x = 1 or R = r Maximum power transfer

theorem. That’s why there are several output

sockets on the back of a stereo amplifier - so its resistance can be matched to that of the speakers.

x = R/r

P

1

Pmax

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Impedance

The maximum power transfer theorem is an example of impedance matching.

Any medium through which energy is transferred has a certain resistance to the flow - an impedance.

It turns out that for any system involving a transfer of energy from a supplier to a receiver we need the impedance of the supplier and receiver to be equal in order to transfer the max. energy.

Thus, we can consider the impedance of a wire or a string or an ear, etc.

Impedance matching is a common problem in transport of electrical signals

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Measuring Instruments

Analogue meters comprise a coil of wire mounted on a pivot between magnets.

Current passing through the coil causes a deflection of the needle.

The basic moving-coil meter is the D’Arsonval galvanometer.

A current of ~ 1mA gives a full scale deflection (fsd).

Internal resistance of meter is the meter resistance (RM).

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Ammeters and Voltmeters

An ammeter uses a resistive shunt to bypass a known fraction of the current (e.g. 999 mA).

A voltmeter uses a series resistance to extend the measurement range.

A known fraction of the voltage is dropped across the resistance.

For an ideal ammeter: RM 0 For an ideal voltmeter: RM

Rs

RM

++ --

IM

Is

A

Rs

RM

++ --

VI

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Designing an Ammeter

Worked Example A galvanometer of resistance 75

has a full scale deflection of 1.5 mA. Design a meter to measure 1A at fsd.

IM = 1.5 mA

Is = 1.0 - 0.0015 = 0.9985 A

VM = IM RM = Is Rs

Rs = IM RM / Is = (0.0015 x 75) / 0.9985

Rs = 0.113

Rs

RM

++ --

IM

Is

A1 A

VM

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Designing a Voltmeter

Worked Example Design a meter to measure 25V at

fsd using the same galvanometer.

V = Vs + VM = I Rs + I RM

Rs = (25 / 0.0015) - 75

Rs = 16 591

Rs

RM

++ --

VI

V

Vs Vm

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RC Circuits & Time Constants

At d.c. capacitors are an open circuit

I.e there is no electrical path. The plates of a capacitor will

charge or discharge if the current varies with time.

The rate at which this happens depends upon the series resistance of the circuit and the size of the capacitor.

The series resistance limits the current flowing into the capacitor.

The characteristic time is called the time constant of the circuit.

Units: . F

= R C

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RC Time Constants

+-

R

+- C I

q

t

C C (1-1/e)

R C

ChargingCharging

q

t

I0

I0 / e

R C

DischargingDischarging

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RC Circuits

Around the loop:

and

The time constant property of RC circuits is essential in time-dependent circuits, e.g. oscillators & filters

0Cq

IR

0Cq

dtdqR

Cq

RCRCq

Rdtdq 1

RCt

eCq 1

RCt

eIdtdq

I

0

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Electricity in the Home

What is a fatal current?

Why does house wiring have 3 wires?

How do fuses work?

What is a circuit breaker?

What is an ELCB?

What current can be drawn from power points?

What is an “off-peak” system?

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What is a Fatal Current?

DEATHDEATH

1.0

0.2

0.1

0.01

0.001

Am

pere

s

Painful

Can’t let go

Muscular paralysis

Extreme breathing difficulty

Mild sensation

Threshold of sensation

V = 240 VV = 240 V

R = 1.5 kR = 1.5 k

R = 0.5 MR = 0.5 M

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Why does House Wiring have 3 Wires?

The three wires are live (hot), neutral and earth.

Actually, only two wires come into your house - live & neutral.

Live is the high potential side of the transformer while the neutral is connected to ground at the transformer.

But - neutral may be at a different potential to earth by the time it gets to your house!

The earth wire is the local earth (water pipe, earth stake).

All electrical devices in a metal case have the case connected to earth.

Ensures that if the live wire touches the case then the least resistant path to earth is through the earth wire & not through you!

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How do Fuses Work?

Fuse is a small metallic strip designed to melt when the current exceeds a certain value.

Fuse wire in Woolies rated at 8 A & 16 A for example.

But, bear in mind that plain fuse wire does take a finite time to melt.

In some cases, this means that the wire has time to pass a much higher value of current than its rating!

Special fuses are available - quick blow fuses have a spring that applies tension to the fuse wire - if it starts to melt it is pulled thinner and blows quickly.

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What is a Circuit Breaker?What is an ELCB ?

More modern homes have the fuses replaced by a circuit breaker (CB).

When the current exceeds a certain value the CB acts as a switch & opens the circuit.

A common design involves the use of a bimetallic strip.

When the current exceeds a certain value the strip heats up and bends.

The bending strip breaks the circuit. Can be slow - many CB’s now

incorporate electromagnets. An earth leakage circuit breaker

(ELCB) is a device that detects very small currents (mA) to ground.

If a current is detected then the power is switched off in a few ms.

Could save your life!

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What Current can be Drawn from Power Points?

Its important to be able to calculate the max current that can be drawn.

Typically, a power circuit is fused at 16 A in Australia.

Light circuits are fused at 8 A. Thus, for a single circuit the total

current load must not exceed 16 A.

But - most appliances quote the power drawn and not the current.

Just need to remember that P = IV and that mains voltage is 240 V.

Max power load on a single circuit is:P = 16 x 240 = 3.8 kW

In many old houses in Newcastle all of the sockets in the house are on a single circuit!!!

Be careful when turning stuff on! - especially in winter!

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What is an “off - peak” system?

Demand for electricity is not spread out evenly during the day or year.

This presents problems for the power supply companies and the

management of the power distribution network.

To encourage more even use of power the cost of electricity

supplied during low demand periods “off peak” is reduced.

This usually occurs after 11pm and is measured by a separate meter

box. The meter box is activated by a high

frequency signal transmitted down the power cable to your house.

Usually operates water heaters and household storage heaters.

Off-peak power is also used to store energy - e.g. hydro-systems.