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PHYS113 Electricity
and Electromagnetism
Semester 2; 2002
Set B Notes (WK-10
final) Professor B. J.
Fraser
1
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 produced
Precipitators (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.
2
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.
charge on either
Capacitance =
p.d. between them

Q
C
V


Unit = Farad, F, = C/V
Large unit, - usually use mF, 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.
3
Parallel Plate Capacitor
+Q
Area, A



-
+ +
- - -Q
+ +
-- +++ - + +
d
From Gauss’ law,
Q
Q
2 EA   E 
for each plate:
0
2 0 A
E
For 2 || plates:
Q
0 A
But electric field, E = V/d,
Qd
0 A
Q
V 
C  Q
0 A
V
Qd
C
0 A
d
4
Cylindrical Capacitor
+Q
2a
2b
-Q
l

Inner conductor radius = a, Inner
conductor radius = b, length = l.
b
Vb  Va    E  d s
a
2k
 From Gauss’ law,
E
r
b
b
dr
b

V    Er dr  2k   2k ln 
r
a
a
a

Thus: C 
Q
C 
V
Q
Q

V 2kQ  b 
ln 
l
a
l
b

2k ln 
a
Important for
practical capacitors
& shielded cables
5
Capacitance of an Isolated
Sphere


Potential of a sphere is:
Q
V
4 0 r
Thus, the capacitance is:
Q
C   4 0 r
V
Example:
 Capacitance of the Earth is:
7.1 x 10-4 F
6
Energy in Capacitors





The electric field contains energy
q
Work to move
dW  Vdq  dq
C
a charge dq is:
For total charge, Q:
Q
Q
q
1
Q2
W   dW   dq   q dq 
C
C0
2C
0
The work is stored as potential
energy in the electric field:
Q 2 CV 2 QV
U


2C
2
2
Since electric field is:
E = V/d
0E 2
CV 2  0 A 2 2  0 E 2
U

E d 
 Ad  
volume 
2
2d
2
2
energy
Energy density =
volume
0E 2
U
u

i.e. energy  E2
volume
2
7
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.
0E 2
u
J m-3
2
E
2u
0
2 1

-12
8.85 x 10
5
-1
 4.75 x 10 N C
I.e. Large !
8
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:

Ctotal
Qtotal

V
Q1  Q2

V
 Ceq  C1  C2
Q1 +
+
- C1
-
Q2
C2
+
-
9
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.
Q Q Q

 
Ceq C1 C2
1
1 1

 
Ceq C1 C2
V1
+
V2
-
+
C1
C2
+
-
10
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/k,
where k is a constant.
11
Dielectric Constant




The capacitance therefore increases
as well:
C = k C0
where k = dielectric constant
This allows capacitors to be made
smaller by using high k dielectrics.
The energy density in the electric
field is also reduced to:
u = u0 / k


Arises because it takes work to
insert the dielectric
Piezo-electricity.
12
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,
Q
I av 
average current:
t







dQ
and at any time,
I
dt
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.
13
Currents in Materials
(or, Why Do Lights Turn On?)

Current comprises charges flowing
across an area dA at velocity v:
dA
I   J  d A
A
v





where J = current density = I/A for
small A.
The number of charges passing
through A is n x A.
Q
I 
 nq vd A
t
And:
J = n q vd
where vd = drift velocity of charges
and: n = number density.
14
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.

I
 I  nq vd A  vd 
nqA
What is n?
number of electrons
n
unit density
N A  Cu

M Cu

6.02 x 10 8.92

23
63.5
 8.5 x 10 22 electrons cm -3
 8.5 x 10 28 electrons m -3
15
Current in a Wire

A
= cross-sectional area of wire
=  r2
=  (5 x 10-4) 2
I
vd 
nqA
I

nqr 2
0.5

8.5 x 1028 1.6 x 10-19  2.5 x 10-7 
 4.7 x 10-7 ms -1


Thus, it takes 6 hours for an
electron to move 1 m.
Why do the lights turn on so quickly?
16
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 = s E
where s = 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 / l
A
E = J / s
J 
I sV

A
l
vd
l
17
Conductivity of Materials


1  l 
l


V     I     I
 s  A 
 A
where  = resistivity = 1 / s
Thus:
V=IR
where
R =  l/A = resistance
 Unit = Volt / Amp:
 = V / A
I
I

Ohmic conductor
(e.g. resistor)



V
V
Non-ohmic conductor
(e.g. diode)
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 .
18
Electric Power
or, 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:
P



dW
dq
V
Dt
dt
But I = dq/dt: P = VI
Unit = Watt,
1W = 1 J/s
For an ohmic material the power
dissipated (mostly in heat) is:
2
V
P  IV  I 2 R 
R
19
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
20
7. Direct Current Circuits
Sources 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:
dW

dq

Unit is the Volt (= J/C)
21
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
I
+

-

R
The net potential energy around the
circuit is:
- I R = 0
22
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
Rint
+
-

R
V =  - I Rint
23
Sources of EMF
Name
Converts
Battery
Electrical
Chemical 
Generator
Electrical
Mechanical 
Solar Panel
Radiation  Electrical
Thermocouple
Electrical
Heat 
MHD
Magnetic  Electrical


d
r b
R around
Potential
a circuit:
c
I
+
a'
-
a
Ir
IR
V
a a' b c d
24
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

V1
I
V2
I
R1
+
R2
I
V


Around the circuit loop.
V = I (R1 + R2)
Therefore:
Req = R1 + R2
25
Resistors in Circuits
Resistors in parallel
 Same p.d. across each resistor.
 Current is shared between resistors.
I1
1 1
I  I1  I 2  V   
 R1 R2 
R1
I2
I


R2
V
+
1
1 1
 
Req R1 R2
-
In household circuits appliances are
connected in parallel.
Xmas tree lights are often in series.
26
Circuit Analysis:
Kirchoff’s Laws
Complex circuits involving multiple
loops are analysed using Kirchoff’s
Laws.
First Law: At a Junction
 The sum of the currents entering
and leaving the junction is zero.
 Statement of conservation of charge

I
in
  I out
Second Law: Around a Circuit Loop
 The sum of potential changes is zero
 The potential is conserved.
 Statement of conservation of energy
 V  0
loop
27
Circuit Analysis Example
Worked Example
 Find the current in each branch of
the circuit shown below.
2
+ 10 V
-
3
1
+ 5V
-
4
10 
28
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
I1
1
I3
I2
10 
I1 + I2 = I3
(1)
29
Circuit Analysis Solution

Sum potential drops around
first loop.
Start here
I1
-
2
+
- 1 +
+ 10 V
a
-
3
+
+ 5V
- - 4 +
I2


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
 - 5 I1 + 5 I2 - 5 = 0
 I1 - I2 = 1
(2)
30
Circuit Analysis Solution

Sum around the other loop.
+ 10 V
- - 3 +
- 2 +
I1
I3
+
10 
-
10 - 2 I1 -10 I3 - 3 I1 = 0
 - 5 I1 - 10 I3 + 10 = 0
 I1 + 2 I3 = 2

(3)
Solve simultaneously & check !
I1 = 0.8 A,
I2 = - 0.2 A,
I3 = 0.6 A
31
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:
Rr

the power delivered to the load is:
P  I 2R 

R
-

Rint
+
I

I
2
R  r 
2
R
When is P a maximum?
32
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!
P
Pmax
1



x = R/r
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.
33
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
34
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).
35
Ammeters and Voltmeters

An ammeter uses a resistive shunt to
bypass a known fraction of the
current (e.g. 999 mA).
Is
Rs
+
-
IM


RM
A voltmeter uses a series resistance
to extend the measurement range.
A known fraction of the voltage is
dropped across the resistance.
Rs
+
I


A
RM
V
For an ideal ammeter: RM  0
For an ideal voltmeter: RM  
36
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.
VM
Is
1A
Rs
+
IM
RM
A
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 
37
Designing a Voltmeter
Worked Example
 Design a meter to measure 25V at
fsd using the same galvanometer.
Vs
Vm
Rs
+
I
RM
V
V
V = Vs + VM = I Rs + I RM
Rs = (25 / 0.0015) - 75
Rs = 16 591 
38
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.
t=R C

Units:
.F
39
RC Time Constants
R
I
+ -
+
-

C
q
C
C  (1-1/e)
Charging
RC
q
t
I0
Discharging
I0 / e
RC
t
40
RC Circuits

Around the loop:
q
  IR   0
C
dq q
  R   0
dt C
dq 
q
1
q  C 

 

dt R RC
RC
t

 q  C 1  e RC 




and
t
dq
I
 I 0 e RC
dt
The time constant property of RC
circuits is essential in timedependent circuits, e.g. oscillators &
filters
41
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?
42
What is a Fatal Current?
1.0
0.2
Amperes
0.1
0.01
V = 240 V
DEATH
R = 1.5 k
Extreme breathing
difficulty
Muscular paralysis
Can’t let go
Painful
Mild sensation
R = 0.5 M
Threshold of sensation
0.001
43
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!
44
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.
45
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!
46
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!
47
What is an “off - peak” system?

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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.
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