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AS – Current Electricity
Textbook chapters 4-5
Chapter 4 – Back to Basics
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Charge
Current
pd
energy
power
resistance
resistivity
What is “electric charge”?
• This is a surprisingly difficult question to answer!
– It’s like being asked “what is mass?” or “what is time?”
– Electric charge is a fundamental property of matter
• It was first encountered through static electricity:
– Friction between two, usually non-conducting, materials
sometimes results in attractive or repulsive forces (suggesting
two “flavours”)
– They are then said to be charged, and the logic of how the
charge was brought into being suggests the law of charges:
– Like charges repel
– Unlike charges attract
•
and we note a third feature:
– The force decreases in strength as separation increases
Charge on the move
• Sparks and other effects suggest that charge can
move.
– and does so to neutralise accumulated charge
• The earth can act as an infinite source or sink for
charge
• Conductors allow charges to flow through them and
insulators do not, seeming to leave the charge
“static”.
• Conductors can only become charged if they are
insulated from their surroundings, e.g., suspended by
nylon threads.
Conductors and insulators
• Generally, most metals are good
conductors and most non-metals are not.
• An electric cable makes use of both types
of material.
Metal wires to
carry electricity
Plastic insulation to
isolate metal wires
• There is a third category, semi-conductors,
which are also very useful
– More charge carriers released if energy in
Electric current
• An electric current is a flow of electric
charge.
• Current is measured in Amperes
(Amps, A)
• In metals, the charges that flow are
negatively charged electrons.
• Some electrons in metals are
essentially “free” from their parent
atoms, so they can move within the
metal.
– This makes metals good conductors.
– Insulators do not have many free charges
The Nature of Charge
• So, It is known that in solids, negative
charges are mobile and associated with
discrete particles (electrons) of a particular
mass and charge (-1.6 × 10-19 coulombs).
• Except in fluids, the positive charges remain
static.
• Charge is hard to pin down and measure.
• Current is easier to get a grip on – it has
various properties and can exert forces (the
motor effect) which allow a base unit, (the
ampère), to be defined.
Charge and Current
• The charge flowing in a given time (in
seconds) is then defined in turn as
Q  It
charge (C)
current (A)
time (s)
Q
so I 
t
• One coulomb is the charge flowing past a
point in one second in a current of 1 amp.
– How many electrons do you need to have a
charge of 1C?
Measuring current
AMMETER
An ideal ammeter
does not affect the
circuit, no energy
is transferred to it:
it has zero
resistance
• Ammeters need to go in the circuit so the current
can flow in and out of them.
• To just indicate the flow of current without
measuring it we can use an indicator light.
Current and Charge
• A graph of current against time is needed if the current flow
is not constant. The area under the graph between two
moments in time represents the charge that has passed.
Find the total charge
delivered from a car battery if
the variation in current
supplied is given by the
adjacent graph.
Charge Carriers
• Each charge carrier takes a time, t, to cross a certain length of
conductor, at a typical speed v. Length therefore is vt.
• The conductor volume is Area of Cross-section, A, × its length
or
Avt
• There are n charge carriers per unit volume of the conductor,
each carrying charge q. In total there is a moving charge of:
n × Avt × q
• Current is I = total charge / time = nAvtq /t = nAvq
Charge Carriers: I = nAvq
• n is the number of charge carriers per unit volume of the
conductor
• v is an average drift speed of the carriers.
• A is the cross-sectional area of the conductor
• q is the charge carried by each carrier (often this is
1.6 × 10-19 C because the carrier is an electron)
Of these, A and q can be measured as can the current flowing
in a given situation. The charge carrier density, n, has to be
estimated, for example, 1 free electron per atom of copper in a
wire. In this way the drift speed of the carriers can be estimated.
Worked Example
• A copper wire of diameter 1.4 mm connects to a filament of a
light bulb of diameter 0.02 mm. A current of 0.42 A flows
through both of the wires.
• Copper has 8 x 1028 free electrons/m3, tungsten 3.4 x 1028 /m3.
For copper, v
= I/(nAq) =0.42/(8x1028 × π × (0.07 × 10-3)2 × 1.6 × 10-19)
=
For tungsten,v
2.13 × 10-5 m s-1 or 0.02 mm s-1
= 0.42 / (3.4 × 1028 × π × (0.01 x 10-3)2 × 1.6 × 10-19)
=
0.246 m s-1 or 246 mm s-1
In tungsten there are fewer charge carriers /m3, moving faster not slower, colliding more
vigorously with the lattice atoms, losing energy as heat in the conductor.
So, although the electric signal travels at close to the speed of
light, the charge carriersare going much more slowly
How can this be?
Electric current
• Where do the electrons come from?
– In a circuit, all the wires and components are
full of electrons, so as soon as a power supply
is connected a current starts flowing
– No time delay with long wires!
Remember the sign
convention:
– Remember bicycle chain model
SQ p.47
When you pedal, all links move at once
Real Current flow
Current will therefore
not be an orderly flow of
electrons more a drifting
superimposed on the
randomly moving
“electron gas”
Think of a swarm of
midges being carried
sideways by an air
current.
How fast?
• Coloured ions in solution in filter paper show a
very slow progress in between two electrodes.
• Calculations using I = nAvq assuming each metal
atom releases one electron give speeds of a few
millimetres per second for normal currents.
• Think of a vast sluggish river rather than a fast
white-water rafting stream. Both can deliver a
similar volume of water per second despite
different speeds.
Speeds to remember
• Signal speed – speed of light
• Thermal speed – 100 m/s
• Drift speed – 1mm/s
• Make sure you understand this!
Electrical Energy
• Electricity is useful because it can be easily
converted into other types of energy.
• A battery or power supply gives electrical
energy to the electrons in a circuit.
• Other circuit components then convert this
to different forms of energy.
Potential Difference (“Voltage”)
• The p.d. between two points is a measure of
how much electrical potential is transferred to
or from the charges as they pass between
those points.
• The p.d. of a power supply is a measure of
how much electrical potential is transferred to
or from the charges as they travel between its
terminals
– This can be thought of as how hard it “pushes”
charge
– so, for any circuit, a larger voltage means a larger
current flowing
What is electrical potential?
• “Potential” is the electrical potential energy
per unit charge
– so if more charge flows between two points more
energy is transferred, but the p.d. between those
points is fixed.
– Unit of potential?
– J/C (or V)
• Note: charges are not “used up”
– Energy is transferred
– The bigger the p.d., the more energy is
transferred
Potential difference and energy
• Potential difference is the work done (or
energy transferred) per unit charge
W
V
Q
work (J)
charge (C)
pd (V)
• Separating charges requires an input of
energy.
– Force is needed to overcome the attraction of opposite
charges and work has to be done to move them.
– Pushing a number of like charges together against their
natural repulsion also requires an active force and work to
be done.
Measuring voltage
• To measure the voltage between two
points we connect a voltmeter in parallel
across those two points
Here we are measuring
the voltage across the
resistor R
An ideal voltmeter does not
affect the circuit, no energy
passes through it: it has infinite
resistance
Energy, work done and power
• Work is done when energy is transferred
from one form to another
– Electrical to...?
• W  QV , Q  It
so W  ItV
work done (J)
W ItV
 IV
• P 
t
t
power (W)
SQ p. 49
Examples 1
Resistance
• Resistance is the opposition to
current flow displayed by
components
– for a fixed voltage, the larger the
resistance, the smaller the current
• Resistance of connecting wires is
usually so small it is ignored
• Resistors dissipating energy get hot!
– e.g. lamp filament
Resistance
• Resistance is caused by collisions between the
free charges and the lattice of atoms which
makes up the conductor
• Each collision transfers energy to the atoms of
material – material heats up
• A high current means more collisions – resistor
gets hotter
Resistance model
Resistance
• Resistance is a measure of the opposition
to current flow
• We define the resistance of a device as the
voltage needed to push a given current
V
through it
voltage (V)
R
resistance (W)
(Ohms)
I
current (A)
• When a p.d. of 1V causes a current of 1A to
flow through a device, its resistance is 1 W
Voltage, Resistance & Current
• We have
resistance (W)
V
R
I
voltage (V)
current (A)
• So for a given circuit:
– What happens to the current if we increase the
voltage of the power supply?
– What happens to the current if we increase the
resistance of the components?
Examples
• If a lamp has a current of 3A when there is
a p.d. of 12V across it, what is the
resistance of the lamp?
• What is the current through a 100W
resistor with a p.d. of 5V across it?
• A real ammeter has a resistance of 0.5W.
What will the p.d drop across it be when a
current of 5A is flowing?
How can we investigate
resistance?
• Draw a circuit which
would allow us to
measure the
resistance of a
component.
– What quantities do
we need to measure?
– How will we use
them?
– What factors do we
need to control?
device under test
V
• R
I
How can we investigate
resistance?
• So measure V and I,
calculate R
• By adjusting the
variable resistor we
can change V and
see how I changes
– What might you
expect?
– It depends what the
device is...
device under test
V
?
I
Let’s measure some things
• We can use a data logger to speed things
up
• Is there a better way of varying the p.d.?
– What is the limitation of this method:
device under test
A better way to vary p.d.
• A potential divider
• The full range of the supply
pd is accessible, down to
zero
A
• Obtain IV curves for
– a resistor
– a lamp
– a thermistor
– a diode
Vbattery
V
I-V Graph for a wire/resistor
Flash simulation
• For a wire, we find:
– Current is proportional to
voltage
• We find the resistance by
calculating V/I for a point(s)
on the line
• Proportional means I
increases at the same rate as
V, so R is a constant value
– A special case
– Also true for resistors, but not
for all components
I
I2
I1
V1 V2
V
R
I
V1 V2
R 
I1 I 2
V
Ohm’s Law
• Metals and resistors usually obey Ohm’s Law:
“The current through a conductor is proportional to the
voltage across it, provided the temperature remains
constant”
• Not generally true, note the bold type
• Take care if a high current leads to heating:
I
No longer obeys Ohm’s Law
V
Why does resistance
increase with temperature?
Non-Ohmic Conductors
• Lots of components
don’t follow Ohm’s
Law
• e.g. Filament bulb
– The filament is
designed to get hot
and glow
– Its resistance
increases with
temperature
Same behaviour for
reversed current
Thermistor
Flash simulation
• A semiconductor device
• Resistance decreases as it gets hotter
• (Extra energy releases more charge carriers)
Resistance and Temperature
• The resistance of a metal increases with
temperature – positive ions vibrate more and
scatter the conduction electrons so they do not
pass as easily
– This is a positive temperature coefficient
• The resistance of an intrinsic semi-conductor
decreases with increasing temperature because
the number of charge carriers increases
– The % change is much more than for a metal
– Can be used in a temperature sensor
Diodes
• A “one way valve” for
electricity.
– useful to protect circuitry,
covert AC to DC
• Low resistance to +V
(after a small threshold
~0.6V)
• High resistance to –V (up
to a maximum voltage)
• Can be light-emitting when
conducting (LED)
0.6V
Resistivity
• What factors determine the resistance of an
object?
– material
– size
– dimensions
• Different-sized or -shaped lumps of the same
material will have different resistances
• Wouldn’t it be nice to be able to characterise
a material with a single measure?
– resistivity
Resistivity
• resistance of a conductor is
– proportional to its length, L
– inversely proportional to its cross-sectional
area, A
– proportional to the resistivity, r, a material
constant resistance (W)
• R  rL , so r  RA area (m2)
A
L
resistivity (Wm)
length (m)
experiment
Superconductors
• Research exercise:
– What are they?
– Why does superconductivity happen?
– What is special about superconducting
magnets?
– Describe and explain two uses of
superconductors
Superconductivity
• A superconductor has zero resistivity at and below a critical
temperature (TC) that is material dependent.
• A current passing through a superconductor experiences no
drop in p.d., and the current has no heating effect.
• High temperature superconductors have TC above 77 K
(–196°C the boiling point of liquid nitrogen)
• Used in:
– high power electromagnets (MRI scanner, CERN) generating
strong magnetic fields,
– power cables that do not waste energy
Circuit symbols
• You’ve just got
to learn them!
Chapter 5 – DC circuits
• Kirchhoff’s Rules
(1845) are two
simple statements of
conservation laws
which allow us to do
circuit calculations.
Gustav Kirchhoff (1824-87)
Kirchhoff’s current rule
• At any junction in a circuit the total current
leaving the junction equals the total
current entering the junction
– i.e. charge is conserved
– So components in series must have the same
current flowing through them
Electric current
• When a circuit with a battery is competed,
the battery “pushes” the charges around.
• Electric current is not “used up” as
electrons flow around a circuit.
– The current is the same through all
components in series circuit
– All ammeters read the same:
– (note
A
connect in series)
Current in a parallel circuit
• The total current through the battery is
equal to the sum of the currents through
each parallel branch
I battery  I1  I 2  I 3
• The smallest current flows
through the branch with the
highest resistance.
Ibattery
I1
I2
I3
Kirchhoff’s voltage rule
• The net voltage drop around any closed
loop path is zero.
– i.e. energy is conserved
– For multiple components in series, the total pd
across all the components is equal to the sum
of the potential differences across each
component
Kirchhoff’s voltage rule
• The net voltage drop around any closed
loop path is zero.
– i.e. energy is conserved
– The pd across components in parallel is the
same
Try SQs on p. 60
Voltage in a series circuit
• The energy transferred to the charge by the
battery = the energy dissipated by all the
components in the circuit
Vbattery  V1  V2  V3
• The largest resistance has
the largest voltage across it
(most energy transferred)
Vbattery
V1
V2
– If all resistances are equal, the battery voltage is
divided equally
V3
Voltage in a parallel circuit
• All components connected to a battery in
parallel have the same voltage across
them.
Vbattery  V1  V2  V3
• The current through each
component is the same as if
the other components weren’t
there.
Vbattery
A
Resistors in series
• If you have resistors in series, the current
has to flow through them all
– so the total resistance is larger than any
resistor on its own...
VT  V1  V2  V3  ...
IRT  IR1  IR2  IR3  ...
VT
R1
R2
R3
RT  R1  R2  R3  ...
Resistors in parallel
• If you have resistors in parallel, the current
splits so some flows through each resistance
– so the total resistance is smaller than any
resistor on its own...
I T  I1  I 2
R1
R2
V
V V
 
RT R1 R2
1
1
1
 
 ...
RT R1 R2
Resistance is “getting in the way”
• The heating effect of a current is due to
charge carriers repeatedly colliding with the
positive ions of the component.
• After losing Ek the charge carrier is
accelerated again by the p.d. across the
material – until it collides again
• There is a net transfer of energy from the
current to the positive ions as a result, i.e.
“the current does work against the
resistance”.
Resistance is
hot work
• The pd in a mains circuit is constant and we can choose
the resistance of a component, so in terms of V and R, I =
V/R
P = IV = V2/R or (I2R, but varying R changes I)
• Highest rate of energy transfer occurs when R is lowest!
(but higher than the resistance of the mains cables –
why?)
• Energy transferred over time = I2Rt or V2t/R
Try SQs on p. 63
• Rate of heat transfer is P = I2R or V2/R
• the hotter a component is, the faster it
will lose heat to the surroundings.
• maintaining a temperature is a balancing
act; when rate of heat produced = rate of
heat loss, the temperature remains
constant
• Both AC and DC
have the same
heating effect.
Resistance is futile, Mr Bond
Model Resistance
Real Cells have resistance of their
own
• At the same time that the chemical
reaction in a cell, or the electromagnetic
interactions in a generator are producing a
pd, they also oppose the flow of current
• as soon as the current starts to flow, it
experiences resistance within the cell or
generator itself.
• Internal Resistance
emf
• The electromotive force [ emf, ε] of a
source is the energy per unit charge
produced by the source.
• ε = Energy /Charge = E/Q
Terminal pd
• The PD across the terminals of the
source is the electrical energy per unit
charge delivered by the source when it is
in a circuit.
• Terminal pd is less than the emf, some
“volts get lost” overcoming the internal
resistance of the source.
The algebra ...
• ε = Ir +IR
• IR is the terminal PD
measured across the load
resistor, V
• So Ir is the PD lost inside
the cell overcoming internal
resistance
• and Ir = ε - V
• r = (ε – V)/I
• “The internal resistance of a
source is the loss of PD per
unit current in the source
when current passes
through the source”
Measuring internal resistance
• As we vary the circuit
resistance we can
measure how current
and terminal pd
change
• (The lamp limits the
current)
Measuring internal resistance
e  IR  Ir
e  V  Ir
V   rI  e
( y  mx  c)
• So a plot of V against I
should give a straight
line
• Gradient is -r
• y intercept is e
Calculating internal resistance
• If we know two values of I and V we can
calculate r:
V1  e  I1r
V2  e  I 2 r
so V1  V2  e  I1r  (e  I 2 r )  r ( I 2  I1 )
V1  V2
and r 
I 2  I1
Try SQs on p. 66
When to worry about internal
resistance?
• When answering A level questions!
• If R>>r, then internal resistance becomes
negligible and terminal pd  emf
• Safest to include it, unless the question
says you can ignore it
Cells in series
• Emfs add
• Internal resistances add
• Same current flows
• Watch out for reversed cells!
Cells in parallel
• For identical cells:
– I/n flows through each
– So lost pd is Ir/n for each
– So terminal pd V=e-Ir/n
• so the source has an emf of e and internal
resistance of r/n
• Cells in parallel can deliver more current
than a single cell
Arrays of cells
• e.g. solar panels
• Connect in rows to give
the pd required
• Connect rows in parallel to
provide a suitable current.
Try SQs on p. 69
Circuits with diodes
• Simple Model: a
silicon diode
– once it starts
conducting in forward
bias
– maintains a pd across
its terminals of 0.6V
– whatever current
passes through it
– in reverse bias it has
infinite resistance.
SQ p. 69
The Potential Divider
• Essentially a pair of
resistors, used to split the
applied potential difference
as desired.
V0
V0
I

RT R1  R2
 R1 
V0
V1  IR1  
 R1  R2 
 R2 
V0
V2  IR2  
 R1  R2 
The Potential Divider
• Varies resistance from 0 to
a maximum
• Allows the rest of the circuit
maximum current down to a
minimum value, but not
zero current
• varies pd at the slider
from 0 to maximum
• can feed the rest of the
circuit any pd/current
from 0 to maximum
• always draws current
through itself
Variable resistor
Potential Divider
The Potential Divider
Variable PD
Fixed PD
Sensor Circuits: using LDRs
Light increases Vout
Light decreases Vout
Uses of potential dividers
SQ p. 71