Module: Batteries and Bulbs Time allocation: 10 hours

1
Module Batteries and Bulbs Time allocation 10
hours
IJSO Training Course Phase II
2
Objectives

• Introduce a model of electrical conduction in a
  metal, and the concepts of resistance and
  internal resistance.
• Define and apply the concepts of current, and the
  use of ammeters and voltmeters.
• Draw circuit diagrams with accepted circuit
  symbols.

3
1. Electrical Conduction in Metals

• A solid piece of metal, at room temperature,
  consists of metal ions arranged in a regular
  pattern called a crystal lattice and free
  electrons moving in the spaces between the ions.
• The motion of the free electrons is random. We
  say they have random thermal motion with an
  average speed which increases with temperature.

4

• The figure below represents a piece of metal
  which does not have current flowing through it.
  The arrows represent the random thermal motion of
  the electrons (their average speed at room
  temperature is hundreds of kms-1).

5

• If a current is flowing in the piece of metal,
  then another motion is added to the random
  thermal motion (see the figure below). This
  motion is more regular and results in a general
  drift of electrons through the metal. A typical
  drift velocity for electrons in metals is less
  than 1mm/s . The magnitude of the drift velocity
  depends on the current, the type of metal and the
  dimensions of the piece of metal.

6

• The resistance of a piece of metal is due to
  collisions between the free electrons and the
  metal ions.

7

• During a collision, some of the kinetic energy
  possessed by the electron can be transferred to
  the ion thus increasing the amplitude of the
  lattice vibrations. Therefore, resistance to the
  flow of current causes the temperature to
  increase or in other words, resistance causes
  electrical energy to be converted into thermal
  energy (internal energy).

8

• At higher temperature, the amplitude of the
  lattice vibrations increases, the collisions
  between the free electrons and the metal ions are
  more often. This suggests that the resistance of
  a piece of metal should increase with
  temperature.

9

• Note Not all materials behave in this way the
  resistance of semi-conductors (e.g. silicon and
  germanium) decreases with temperature.

10
Conductors / Insulators

• Electrical conductors readily conduct electric
  charges, small resistance .
• Electrical insulators conduct electric charges
  poorly, large resistance .
• Examples

Good conductors Poor conductors Good insulators
Metals, carbon moist air, water, human body Rubber, dry air
11
2. Electric Circuits

• When drawing diagrams to represent electric
  circuits, the following symbols are used.

Wires crossing but not connected
Wires crossing and connected
12

• Unless otherwise stated, we assume that
  connecting wires are made of a perfect conductor,
  i.e., no resistance.

Switch Battery A.C. supply

13
Resistor Variable resistor Push button

Filament lamp Voltmeter Ammeter

Transformer Rheostat (variable resistor)

14

• Exercise
• A_________, B _______, C ________
• D _________, E _________
• push button, A.C. supply, rheostat, voltmeter,
  bulb

15
3. Electric Current

• Generally speaking, an electric current is a flow
  of charged particles. For examples, a current in
  a metal is due to the movement of electrons. In a
  conducting solution, the current is due to the
  movement of ions.
• Current is measured using an ammeter. An ammeter
  measures the rate of flow of charge. For
  simplicity, an ammeter gives a reading which is
  proportional to the number of electrons which
  pass through it per second.

16

• The unit of current is the Ampere, A.

An ammeter is always connected in series with other components. The resistance of an ammeter must be low compared with other components in the circuit being investigated.
17
Current in Series Circuits

• A current of 2A corresponds to a certain number
  of electrons flowing in the circuit per second.
  So if I1 2A, I2 and I3 must also be 2A because
  in a series circuit, the electrons have only one
  path to follow.
• Conclusion The current is the same at all points
  in a series circuit.

18

• If the three current I1, I2 and I are measured it
  is found that

I1 I2 I
This result is called Kirchhoffs current law, stated as follows. The total current flowing towards a junction in a circuit is equal to the total current flowing away from that junction.
19

• As an analogy, consider vehicles at a road
  junction.
• The number of vehicles passing point 1, per
  minute, must be equal to the number of vehicles
  passing point 2 per minute plus the number of
  vehicles passing point 3 per minute.

20
Relation between Current, Charge and Time

• Another analogy is often found to be helpful.
  Consider a pipe through which water is flowing.
  If the rate of flow of water through the pipe is,
  for example, 25l min-1, then in 15 minutes, the
  total quantity of water which has moved through
  the pipe is 25 x 15 375l . The quantity of
  liquid is equal to the rate of flow multiplied by
  the time.

21

• Similarly, when considering a flow of electric
  charges, the quantity of charge which passes is
  given by
• the unit of charge is the Coulomb. The Coulomb
  can be now defined as follows

Q I t
quantity of charge rate of flow of charge time
1 C is the quantity of charge which passes any point in a circuit in which a current of 1A flows for 1sec.
22

• It should be noted that the Coulomb is a rather
  large quantity of charge. 1 C is the quantity of
  charge carried by (approximately) 61018
  electrons!
• Hence, each electron carries
• 1.6022  1019 C. This is the basic unit of
  electric charge.

23

• Exercises
• 1. A current of 0.8A flows through a lamp.
  Calculate the quantity of electric charge passing
  through the lamp filament in 15 seconds.
  12 C
• 2. A current of 2.5A passes through a conductor
  for 3 minutes. Calculate the quantity of charge
  passes through the conductor.
  450 C

24
4. Voltage

• When a body is falling through a gravitational
  field, it is losing gravitational potential
  energy. Similarly, when a charge is "falling"
  through an electric field, it is losing electric
  potential energy.
• Water has more gravitational potential energy at
  B than at A so it falls. The potential energy
  lost by 1 kg of water in falling from level B to
  level A is the gravitational potential difference
  (J/kg) between A and B.

25

The flow of water can be maintained using a pump. A flow of electrons can be maintained using a battery. The battery maintains an electrical potential difference between points A and B.
26

• To measure voltage we use a voltmeter. The unit
  of voltage is the volt.
• A voltmeter gives us a reading which indicates
  the amount of energy lost by each Coulomb of
  charge moving between the two points to which the
  voltmeter is connected. 1V means 1 JC-1.

A voltmeter is always connected in parallel with other components. The resistance of a voltmeter must be high compared with other components in the circuit being investigated.
27
What is an ideal voltmeter?

• An ideal voltmeter can measure the potential
  difference across two points in a circuit without
  drawing any current.

28
Voltages in Series Circuits

• Consider the simple series circuit above.
• Energy lost by each Coulomb of charge moving from
  A to B is V1.
• Energy lost by each Coulomb of charge moving from
  B to C is V2.

29

• Energy lost by each Coulomb of charge moving from
  C to D is V3.
• Obviously the total amount of energy lost by each
  Coulomb of charge moving from A to D must be V1
  V2 V3 ( V).
• Conclusion The total voltage across components
  connected in series is the sum of the voltages
  across each component.

30
Voltage across Components in Parallel
All points inside the dotted ellipse on the right must be at the same potential as they are connected by conductors assumed to have negligible resistance. Similarly for all points inside the dotted ellipse on the left. So the three voltmeters are measuring the same voltage.
31

• Conclusion Components connected in parallel with
  each other all have the same voltage.
• Again, this does not depend on what the
  components are.

32
5. Resistance

• The resistance of a conductor is a measure of the
  opposition it offers to the flow of electric
  current. It causes electrical energy to be
  converted into heat.
• The resistance of a conducting wire is given by

R ?l /A
33

• The unit of resistant is Ohms (W). It depends on
  the length of the piece of metal l and the
  cross-sectional area of the piece of metal A, and
  r is called the resistively, units Wm, which
  depends on type of metal.
• In a circuit, the resistance is defined by
• Where V is the voltage across the resistor and I
  is the current flows through it.

Resistant voltage / current
RV/I
34
Ohms law

• The Ohms law states
• As voltage divided by current is resistance, this
  law tells us that the resistance of a piece of
  metal (at constant temperature) is constant.
• Note the resistance of a piece of metal
  increases as its temperature increases.

For a metal conductor at constant temperature, the current flowing through it is directly proportional to the voltage across it.
35

• Exercises
• 1. If the resistance of a wire, of length l and
  uniform across sectional area A, is 10W. What is
  the resistance of another wire made of the same
  material but with dimensions of twice the length
  and triple cross sectional area?
• unchanged

36

• 2. A uniform wire of resistance 4 W is stretched
  to twice its original length. If its volume
  remains unchanged after stretching, what is the
  resistance of the wire? 16 W
• 3. A current of 0.8A flows through a lamp. If the
  resistance of the lamp filament is 1.4W,
  calculate the potential difference across the
  lamp. 1.12 V

37
Effective Resistances

• If two or more resistors are connected to a
  battery, the current which will flow through the
  battery depends on the effective resistance (or
  equivalent resistant), RE, of all the resistors.
  We can consider RE to be the single resistor
  which would take the same amount of current from
  the same battery.

38

• Resistors in Series
• The effective resistance of circuit A is

A B
RE R1 R2 R3
39

• Resistors in Parallel
• The effective resistance of circuit A is

A B
1/RE 1/R1 1/R2 1/R3
40

• Exercise
• 1. A hair dryer consisting of two identical
  heating elements of resistance 70W each is
  connected across the 200V mains supply. The two
  elements can be connected in series or in
  parallel, depending on its setting. Calculate the
  current drawn from the mains in each setting.
  1.4A 5.7A

41
6. Potential Dividers

• In the circuit, let v1 be the voltage across R1
  and v2 the voltage across R2.
• It can be shown that
• Circuits of this type are often called potential
  dividers.

V1/V2 R1/R2
42

• Exercises
• 1. Two resistors are connected in series, show
  that and .
• 2. Two resistors are connected in parallel, show
  that and .
  

43
Variable Resistors

• A variable potential divider can be made using
  all three connections of a variable resistor
  (also be called a rheostat) .
• (i) Rotating variable resistor (internal view)

44

• (ii) Linear variable resistor

45
Using Variable Resistors

• In the circuit below, notice that only two of the
  connections to the variable resistor are used.
• The maximum resistance of the variable resistor
  is 100W.

46

• When the sliding contact, S, is moved to A the
  voltmeter will read 6V (it is connected directly
  to both sides of the supply). This is, of course
  the maximum reading of voltage in this circuit.
• What is the reading of the voltmeter when the
  sliding contact is moved to B?

47

• We have, in effect, the following situation.
• Therefore, the voltmeter will read 3V.

48
Variable Resistor used as a Variable Potential
Divider

• What is the reading of the voltmeter when the
  sliding contact is moved to B?
• The voltmeter reading can be reduced to zero by
  moving the sliding contact to B. The wire "x"
  (assumed to have zero resistance) is in parallel
  with the 100W resistor. This circuit is useful in
  experiments in which we need a variable voltage
  supply.

49

• Exercise 1
• In the circuits above, a variable resistor of
  resistance 100 W is connected to a 50 W resistor
  by means of a sliding which can be moved along
  the variable resistor.
• (a)    Determine the maximum and minimum currents
  delivered by the battery, which has an e.m.f. of
  10 V and negligible internal resistance, in the
  two circuits.
• (b)   Determine also the currents delivered by
  the battery when the sliding contact is at the
  mid-point of the wire in both cases.
• (a) 0.2A, 0.067A 0.3A, 0.1A (b) 0.1A 0.13A

50
7. Electrical Power and Energy

• Any component which possesses resistance will
  convert electrical energy into thermal energy.
• Consider the simple circuit shown below.

51

• The current, I, is a measure of the number of
  Coulombs of charge which pass through the
  resistor per second.
• The voltage, V, is a measure of the number of
  Joules of energy lost by each Coulomb of charge
  passing through the resistor.
• So, the energy per second (power) supply by the
  battery is

P VI
52

• To calculate the power consumed by a resistor
• In series
• In parallel

P I2 R
P V2 / R
53

• Thus electrical energy can be expressed in the
  engineering unit kilowatt-hour (i.e., energy
  dissipated in an appliance of 1 kW rating
  operated for 1 hour)
• Electricity is supplied to our house through the
  mains. The voltage supplied is alternating and
  correspondingly an alternating flow of charges
  occurs in the wire. This kind of electricity is
  called alternating current (a.c.) in contrast to
  the direct current (d.c.) as supplied by a
  battery.

54

• Live wire brown in colour. In Hong Kong, the
  voltage at the live wire changes from 220 V to
  -220V continuously and alternately, so that the
  current flows backwards and forwards round the
  circuit. A switch and a fuse can be installed in
  live wire to prevent the appliance to go 'live'.

55

• Exercises
• 1. An electric cooker with a fuse and a switch in
  series with the heating element is to be
  connected to the pins of a socket. The correct
  connections should be

56

• 2. Three lamps A, B and C of resistance 250 W,
  350 W and 600 W respectively are connected across
  a 200V supply as shown.
• (a) Calculate the potential difference across the
  lamps.
• (b) Calculate the current passing through the
  lamps.
• (c) Calculate the power dissipated in the lamps.
• (d)List the lamps in ascending order of
  brightness.
• a. 200V, 117V, 83V
• b. 0.33A, 0.33A, 0.33A
• c. 27.8W, 38.9W, 66.7W
• d. A, B, C

57
8. Battery and its Internal Resistance

• The metal contacts which are used to connect a
  battery into a circuit are called its terminals.
  For this reason, when the voltage of a battery is
  measured, we often describe the result as the
  terminal potential difference of the battery.

58

• A battery converts chemical energy into
  electrical energy.
• The electrical energy given to each Coulomb of
  charge is called the e.m.f.1, denoted as ,
  of the battery. So the unit of e.m.f is also
  Volt.
• The term "e.m.f." originally came from the phrase
  "electro-motive force". This is now considered an
  inappropriate term as emf is a quantity of energy
  not a force. However, the abbreviation is still
  used.

59

• In the following circuits, the voltmeter is
  assumed to have infinite resistance (a modern
  digital voltmeter has a resistance of around 107
  W). The voltmeter reading is equal to the e.m.f.
  of the battery.

60

• However, the substances of which the battery is
  made have some resistance to the flow of electric
  current. This is called the internal resistance
  of the battery. A more complete symbol to
  represent a battery is shown below.

61

• The resistor, r, represents the internal
  resistance of the battery. The reading of the
  voltmeter across A and B will be
• The terminal potential difference is only equal
  to the e.m.f. of the battery if the current
  flowing through the battery or the internal
  resistant is zero.

V - Ir
62

• Suppose there is an external resistance R in the
  circuit, it can be considered as in series to the
  internal resistance, so we have

I (R r)
63

• Exercises
• 1. A battery of e.m.f. 3 V and internal
  resistance 1.5 W is connected to another battery
  of e.m.f. 3 V and internal resistance 6 W, same
  polarities being wired together as shown in the
  figure. A student says the rate at which
  electrical energy is converted into internal
  energy is zero. Do you agree? Explain briefly.

64

• 2. A cell of e.m.f. 2 V is connected in series
  with a variable resistor of resistance R and an
  ammeter of resistance 0.4 W. By varying R, a
  series of ammeter readings, I, are taken. A graph
  of R against 1/I is then plotted. The value of
  the y-intercept is found to be -3 W. What is the
  internal resistance of the cell?
  2.5 W

65

• 3. A student is given two identical batteries,
  each of e.m.f. 2V and negligible internal
  resistance and two identical resistors, each of
  resistance 4.5 W. Determine the current through
  each resistor in the circuits shown in the figure
  
• (a) 0.444A, (b) 1.78A, (c) 0.222A, (d) 0.889A and
  0.444A

66

• Note Combination of batteries 
• Batteries in series
• - effective e.m.f E1 E2 E3
• - effective internal resistance r1 r2 r3
• Identical batteries in parallel
• - effective e.m.f. E
• - effective internal resistance r/N, where N
  is the number of batteries in parallel.
• - It can supply a current N times larger than
  that can be supplied by one battery alone.
• End