Higher Physics – Unit 2
2.1 Electric fields and resistors
in circuits
Electric Fields
Electric fields exist in the regions around electric charges.
A charge experiences a force in an electric field.
+
+
+
+
+
+
+
+
+
+
-
+
The arrows show the direction a positive charge will be forced to move.
When two opposite charges are close, we see:
+
-
Electric Fields and Conductors
On applying an electric field to a conductor, the free electric charges
(electrons) move.
These free electric charges moving, are called an electric current.
Charges in Electric Fields
For a charge to move through an electric field, work must be done:
work done
(or energy gained
by charge)
(J, Joules)
W QV
potential
difference
(V, Volts)
charge
(C, Coulombs)
The potential difference (V) between two points is a
measure of the work done in moving one coulomb of charge
V
between the points.
W
Q
Example 1
60 J of work is done moving a charge through a p.d. of 2 kV.
Calculate the size of the charge.
W QV
W 60 J
V 2 kV
2 103 V
Q
Q ?
Q
W
V
60
2 103
0.03 C
Example 2
An electron is moved from plate A to plate B as shown.
500 V
-
A
+
B
Calculate the energy gained by the electron.
V 500V
Q electron
1.60 10
19
C
** info from data sheet **
W ?
W QV
1.60 10 -19
W 8 10
17
J
500
Example 3
An electron is moved from rest between two metal plates with a
potential difference of 5 kV.
Calculate the velocity of the electron as it reaches the positive plate.
V 5 kV
5 103 V
Qelectron
1.60 10
W QV
19
C
** info from data sheet **
W ?
1.60 10
W 8 10
16
19
J
5 103
Energy is conserved.
EK
melectron
W
8 10
16
9.11 10
EK
J
31
kg
?
1
2
8 10
16
8 10
16
4.555 10 -31
v
8 10 16
4.555 10 -31
** info from data sheet **
v
1
m v2
2
2
v2
9.11 10-31
1.756 1015
v
1.756 1015
v
41.9 106 ms-1
v2
v2
Example 4
An electron is moved from plate A to plate B as shown.
-
(a)
+
200 V
750 V
A
B
Calculate the energy gained by the electron
V 750- 200
550 V
Q electron 1.60 10 19 C
** info from data sheet **
W ?
W QV
1.60 10 -19
W 8.8 10
17
J
550
(b)
Calculate the velocity of the electron as it reaches plate B.
Energy is conserved.
EK
W
8.8 10
melectron
9.11 10
17
31
EK
J
kg
8.8 10
17
8.8 10
17
** info from data sheet **
v
?
v
2
v2
v
1
m v2
2
1
2
9.11 10-31
4.555 10 -31
8.8 10 17
4.555 10 -31
1.932 1014
1.932 1014
v 13.9 106 ms-1
v2
v2
Definition of a Volt
Consider the equation W Q V.
Rearrange to give V
W
.
Q
This means then that 1 Volt
This is written as:
1 Joule
1 Coulomb
1 V 1 J C -1
Definition of 1 Volt
The potential difference between two
points is 1 volt when 1 Joule of
energy is required to move 1 Coulomb
of charge between the two points.
Circuits with Batteries or Cells
Experiment
Set up the circuit shown.
1.5 V
A
5.6 Ω
Calculation of Current in Circuit
V IR
V 1.5 V
R 5.6 Ω
I
V
R
1.5
5.6
I ?
I 0.27 A
Measurement of Current in Circuit
The reading on the ammeter is
A.
Why are the two results different?
Answer
The results are different because the battery has a resistance inside
it, which we have not taken into consideration previously.
This resistance is called the INTERNAL RESISTANCE of the battery
and has the symbol r.
Internal Resistance
Every electrical source can be thought of as a
source of emf with a resistor in series.
The resistance from within a cell or battery is known as
INTERNAL RESISTANCE.
Batteries or cells with internal resistance should be drawn as
shown.
E
r
Measuring emf of a Battery
Experiment
Connect a voltmeter across the terminals of a battery as shown.
V
r
to measure emf of a battery
ensure no current is drawn
Reading on
V
= emf.
The nominal emf of a battery is usually written on it.
(e.g. an AA battery has an emf of 1.5 V, however checking this with a
voltmeter may give about 1.54 V)
Electromotive Force (emf)
The maximum voltage of a battery is called the electromotive
force (emf) and has the symbol E.
This can be measured when nothing is connected to the battery
(except the voltmeter).
The emf of an electrical supply is:
the number of Joules of
energy given to each Coulomb
of charge passing through
the supply.
Remember: 1 V 1 J C -1
The voltage measured across
the cell is measured between
terminals A and B.
E
r
A
B
This is called the TERMINAL POTENTIAL DIFFERENCE (tpd).
As nothing except a voltmeter is connected to the circuit, this is
known as the OPEN CIRCUIT TERMINAL POTENTIAL
DIFFERENCE.
When the circuit is open, no current is being taken from the source,
and so no energy is lost in overcoming the internal resistance.
This means:
e.m.f. of a source = open circuit t.p.d.
Example 1
A battery of emf 1.5 V and internal resistance 0.5 Ω, is connected to a
6 Ω resistor.
(a)
Calculate the current in the circuit.
Draw diagram and insert values given.
1.5 V
0.5 Ω
6 Ω
1.5 V
0.5 Ω
6 Ω
V 1.5 V
r 0.5 Ω
R 6Ω
Rtot
6.5 Ω
I ?
V
I Rtot
I
V
Rtot
1.5
6.5
I 0.23A
(b)
The 6 Ω resistor is replaced by a 2 Ω resistor.
Calculate the new current in the circuit.
1.5 V
0.5 Ω
2 Ω
V 1.5 V
r 0.5 Ω
R 2Ω
Rtot
2.5 Ω
I ?
V
I Rtot
I
V
Rtot
1.5
2.5
I 0.6 A
(c)
The battery is short circuited using thick copper wire, which
has negligible resistance.
Calculate the short circuit current.
1.5 V
V 1.5 V
Rtot
0.5 Ω
I ?
0.5 Ω
I
V
Rtot
1.5
0.5
I 3A
emf and Internal Resistance
The following circuit is used to find the emf and internal resistance
of a cell.
V
E
must know this
circuit diagram
r
S
A
R
Switch S Open
V
• reading on
V
= emf
• reading on
A
= 0A
r
E
S
A
R
Switch S Closed
• reading on V
falls to a value less than emf.
• the reading now on
V is called the TERMINAL POTENTIAL
DIFFERENCE.
• the tpd is the voltage across the external resistor R.
• voltage across internal resistance of the battery = emf - tpd.
• this voltage is called LOST VOLTS (voltage across internal
resistance).
The ‘lost volts’ increase with current.
This gives us the following relationship:
emf
emf
tpd
lost volts
E IR Ir
(V)
tpd
(V)
In the SQA data book, the equation is given as:
E V Ir
lost volts
(V)
Internal Resistance Problems
Ohm’s Law,
V IR
applies to ALL circuits, as does the fact
that voltages in a series circuit add up to give the supply voltage.
emf = open circuit tpd
E
r
int
[max (total) voltage]
tpd = voltage across Rext
R
lost volts = voltage across rint
ext
Example 1
A cell with emf 2 V and internal resistance 0.75 Ω is connected to an
external resistor of 3 Ω.
(a)
Draw a circuit diagram.
2 V
0.75 Ω
3 Ω
(b)
Calculate the current in the circuit.
E 2V
R 3Ω
r 0.75 Ω
I ?
E V
Ir
E IR
Ir
E I R r
2 I 3 0.75
3.75 I 2
I 0.53A
(c)
Calculate the lost volts.
** Lost volts is due to the internal resistance, so use r. **
I 0.53A
r 0.75 Ω
lost volts ?
lost volts I r
0.53 0.75
lost volts 0.4 V
(d)
Calculate the terminal potential difference.
** tpd is voltage across the external resistor, so use R. **
tpd
I 0.53A
0.53 3
R 3Ω
tpd
IR
?
tpd
An alternate method:
emf
tpd
lost volts
tpd
emf
lost volts
2 0.4
tpd 1.6 V
1.6 V
Measuring emf & Internal Resistance
Experiment (Method 1)
The circuit shown is used to measure the emf and tpd
The lost volts and internal resistance can then be calculated.
V
E
r
S
A
R
Switch S Open
voltmeter reading =
V
emf =
V
I =
Switch S Closed
A
voltmeter reading =
V
tpd =
V
I =
A
Calculation of Lost Volts
lost volts = emf - tpd
=
lost volts =
V
Calculation of Internal Resistance
I
A
r
lost volts
I
emf - tpd
I
r
Ω
Experiment (Method 2)
The following circuit is used to measure internal resistance of a cell.
V
E
r
A
Switch Closed
V
measures the tpd.
A
measures the current I.
R
Record these values in a table of results.
Change the setting on the variable resistor and note the new values of
V and I each time.
Repeat this several times.
Results
tpd (volts)
I (amps)
Graph
Plot a graph of V against I.
voltage/V
current/A
1.6
1.4
Voltage (V)
1.2
1
y = -0.419x + 1.4818
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Current (A)
0.8
1
1.2
Theory
emf
tpd
E V
lost volts
Ir
V E Ir
V
rI E
Compare this with the equation of a straight line:
r
y mx c
gradient of graph
E intercept on V axis
To calculate the gradient of the graph:
The internal resistance of the battery is
gradient
Ω.
V2 V1
I2 I1
Q1.
Adapted from Higher Physics SQP [X069/301]
During an experiment to measure the e.m.f. and internal
resistance of a cell, the following graph is obtained.
(a)
Draw a circuit which could be used to obtain the data for this
graph.
(2)
(b)
(i)
What is the value of the e.m.f of the cell?
(ii)
Calculate the internal resistance of the cell.
(3)
Conservation of Energy
Electromotive Force
When a circuit is open (no current flowing) the pd across the
terminals is in fact the emf.
When a current flows through components, the sum of the energies
produced in each component equals the total energy provided by
the source.
emf of closed
circuit
=
sum of pd’s across
components
Resistors in Series
The energy lost from the electrical source E, is gained by each of the
resistors in series.
energy lost
E
R1
R2
E
R3
I
energy gained
V1 V2 V3
I RT
I R1 I R2 I R3
I RT
I R1
R2
R3
since current same at all
points in a series circuit
RT
R1 R2 R3
** MUST BE ABLE TO DERIVE **
Conservation of Charge
Resistors in Parallel
The current (charge per second) splits up in a parallel circuit.
E
R1
R2
R3
I1
I2
I3
IT
E
R1
R2
R3
I1
I2
I3
IT
E
IT
I1 I2 I3
E
RT
E
R1
1
RT
E
E
R2
1
R1
E
R3
1
R2
1
R3
since voltage across each
resistor constant and equal
to supply
1
RT
1
R1
1
R2
** MUST BE ABLE TO DERIVE **
1
R3
Resistance Calculations
Calculations involving resistors will require you to add resistors,
some in parallel and some in series.
Example 1
The circuit shows resistors
connected as a potential divider.
Calculate the voltmeter reading:
(a) when the switch S is open
(b)when the switch S is closed.
10 V
R1
6Ω
R2
3Ω
6Ω
S
R3
V
(a)
** Switch S open, so have a series circuit with two resistors **
R1
6Ω
R2
6Ω
VT
10 V
RT
R1 R2
RT
6 6
12 Ω
Know total voltage, and total resistance, so can calculate total current.
IT
IT
VT
RT
10
12
0.83 A
** Series circuit,
IT
0.83 A
R
V
6Ω
?
(b)
total current must flow through each resistor. **
V IR
0.83 6
V 5V
** Switch closed, two resistors in parallel now. **
R2
6Ω
R3
3Ω
RP
?
1
RP
1
RP
RP
RP
1
R2
1
R3
1 1
6 3
0.5
1
0.5
2Ω
** Now have two resistors in series. **
R3
R1
6Ω
R1
6Ω
R2
3Ω
6Ω
R2
2Ω
Combined R2 and R3.
R1
R2
VT
6Ω
2Ω
10 V
Now need to combine these
resistors to find total resistance.
RT
R1 R2
RT
6 2
8Ω
Know total voltage, and total resistance, so can calculate total current.
IT
IT
VT
RT
10
8
1.25 A
So can now calculate the size of voltage across R1.
R1
6Ω
I 1.25 A
V1
?
V1
V1
The voltage across R2 is given by:
I R1
1.25 6
7.5 V
10 7.5 2.5 V
Voltage across R3 is also 2.5 V (voltage across components in parallel are same).
Example 2
A potential divider, PQ is set up as shown.
18 V
P
40Ω
X
40Ω
10Ω
Q
Y
Calculate the potential difference across XY.
** Add the two resistors in parallel. **
R1
R2
RP
1
RP
40 Ω
10 Ω
?
1
R1
1
R2
1
1
40 10
1
RP
0.125
RP
8Ω
** Now have two resistors in series, 8Ω and 40Ω . **
18 V
P
40Ω
40Ω
X
10Ω
18 V
Q
Y
P
40Ω
8Ω
Q
RT
RT
8 40
48 Ω
Know total voltage, and total resistance, so can calculate total current.
IT
IT
VT
RT
18
48
0.375 A
So the voltage across 40 Ω is:
R
V IR
0.375 40
V 15 V
40 Ω
I 0.375A
V ?
So the voltage across each of the resistors in parallel is:
18 15 3 V
Question
Calculate the potential difference across XY for the circuit shown.
12 V
P
40Ω
60Ω
X
30Ω
Q
Y
potential difference across XY = 4 V
2007 – Q9
A battery of e.m.f. 24 V and negligible internal resistance is connected
as shown.
The reading on the ammeter is 2·0 A.
The resistance of R is
A
3·0 Ω
B
4·0 Ω
C
10 Ω
D
12 Ω
E
18 Ω
Wheatstone Bridge
A Wheatstone bridge contains four resistors as shown.
R1
R3
R2
V
R4
The bridge is said to be balanced when the reading on V
When this is the case:
R1
R2
R3
R4
= 0 V.
Wheatstone bridge circuits are sometimes drawn:
R1
R2
R1
V
R3
R3
R1
V
V
R4
R2
R2
R3
R4
Take care in numbering the resistors.
Resistors in series are numbered consecutively.
R4
Example 1
( Higher 2000 – A – Q9 )
In the following circuit the reading on the voltmeter is zero.
2.5 Ω
7.5 Ω
V
R
The resistance of resistor R is
0.33 Ω
A
B
0.48 Ω
2.1 Ω
C
D
3.0 Ω
E
27 Ω
9.0 Ω
Balanced Wheatstone bridge =>
R1
R2
R3
R4
R1
R2
R1
2.5 Ω
R2
7.5 Ω
R3
?
R4
9.0 Ω
2.5
7.5
R3
R4
R
9.0
7.5 R 2.5 9.0
R
22.5
7.5
R 3.0 Ω
Question
( Higher 2002 – A – Q9 )
The diagram below shows a balanced Wheatstone bridge where all the
resistors have different values.
Which change(s) would make the
bridge unbalanced?
P
Q
V
R
I.
II.
III.
S
A.
B.
C.
D.
E.
Interchange resistors P and S.
Interchange resistors P and Q
Change the e.m.f. of the battery.
I only
II only
III only
II and III only
I and III only
Consider
R1
3Ω
R2
R3
R4
5Ω
6Ω
10 Ω
values chosen so bridge is
balanced and all resistors have
different values
R1
R2
R3
R4
P
Q
R
S
3
5
6
10
both sides = 0.6
balanced
I – interchange P and S
10
5
6
3
both sides = 2
balanced
II – interchange P and Q
5
3
6
10
both sides ≠ same
unbalanced
Unbalanced Wheatstone Bridge
The voltmeter reading is
non-zero when the
Wheatstone Bridge is
unbalanced.
To calculate voltmeter
reading, use:
V1
R1
R1 R2
VS
Calculate the reading on the
voltmeter (potential
difference between A and B).
6 V
1 kΩ
R1
4 kΩ
R3
A
V
B
2 kΩ
R2
4 kΩ
R4
PD across R1 and R3.
OR
PD across R2 and R4.
Potential Across R2
R1
1 kΩ
R2
Vs
V2
2 kΩ
6V
?
V2
R2
R1 R2
2
2 1
V2
VS
6
4V
Potential Across R4
R3
4 kΩ
R4
4 kΩ
Vs
V4
6V
?
V4
R4
R3 R4
4
4 4
V4
VS
6
3V
potential difference between A and B = 4 - 3
= 1V
Relationship
In a balanced Wheatstone bridge circuit,
V
= 0 V.
If any resistor is changed by ΔR, the bridge becomes unbalanced,
meaning V ≠ 0 V.
We find that:
out of balance voltage
out of balance
voltage (V)
ΔR (Ω)
ΔR
Thermometer
A thermometer can be made using a Wheatstone Bridge circuit.
R1
R3
A
V
R2
R4
B
R3 is a variable resistor.
R4 is a thermistor (its resistance changes with temperature).
The bridge is balanced by adjusting the variable resistor.
When the bridge is balanced,
V
= 0 V.
The scale is calibrated to read °C rather than volts.
As the thermistor is heated or cooled, the bridge becomes unbalanced.
This causes the temperature reading to change accordingly.