Internal Resistance
Electricity & Electronics 3:
Internal Resistance
AIM
When a current flows round a circuit, a potential difference is developed across components
which have resistance. The aim of this unit it so investigate what happens when current flows
through the source, i.e. the power supply.
OBJECTIVES
On completing this unit you should be able to:
• state that an electrical source is equivalent to an emf with a resistor in series, known
as internal resistance.
• describe the principles of a method for measuring the emf and internal resistance of
a source.
• Use graphs of V vs I and R vs 1/I to calculate E and r.
• explain whey the emf of a source is equal to the open circuit pd across the terminals
of the source.
• use the following terms correctly in context:
terminal pd, load resistor, lost volts, short circuit current.
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Electricity and Electronics
Internal Resistance
Voltage Sources
Cells, power supplies and generators are examples of voltage sources.
An ideal voltage source would produce a constant voltage at its terminals regardless of the
current being drawn from the source. This voltage would be equal to the EMF of the source.
a real voltage source does not behave in this way. The terminal voltage is always less than the
EMF and gets smaller and smaller as more and more current is drawn from the source. This
effect is demonstrated nicely when a car is started with the headlights on. The starter motor
draws a large current, and this makes the terminal voltage of the car battery drop so much that
the lights dim!
This happens because no process is 100% efficient - some of the energy per coulomb is
wasted as heat energy. (Remember 1 volt = 1 joule per coulomb). The energy wasted as heat
for each coulomb is usually referred to as ‘lost volts’. They aren’t ‘lost’ as such - they are just
wasted as heat. Energy is still conserved! We call them ‘lost volts’ as they are not available to
the external circuit.
Internal Resistance
A potential drop usually occurs across a resistance. It makes sense to consider that every
voltage source is an EMF with a resistor in series. This is known as internal resistance. The
symbol for a cell would look like:
E
r
The effect of internal resistance
• With no load across the terminals and no current drawn (open circuit)
The terminal potential difference will be equal to the emf E of the cell.
This is because no current is drawn, and so there are no volts lost across r.
• With a load resistor R across the terminals
The terminal potential difference will be less than the emf E.
This is because a current is drawn and flows through the internal resistance r.
The lost volts = Ir.
The voltages associated with a real voltage source
emf (E) - This is the voltage supplied by the cell. It can only be
measured across the terminals when zero current is drawn from the
source.
Terminal potential difference (V) - This is the voltage available across
the terminals of the voltage source when a current is drawn from the
source.
Lost volts (E-V, or Ir) - This is the drop in voltage associated with
driving a current through the voltage supply.
Strathaven Academy
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Electricity and Electronics
Internal Resistance
Solving internal resistance problems
You should start any internal resistance problem with a diagram.
E
r
Remember:
I
lost volts = Ir
V = IR
R
E = IRT
V
The next step is to list the 5 variables involved (remember it as ‘RIVEr’!)
R
I
V
E
r
=
=
=
=
=
External resistance (ie all except internal resistance) (ohms)
Current in circuit (amps)
Terminal potential difference (volts)
EMF of supply (volts)
Internal resistance of supply (ohms)
By applying the law of conservation of energy to this simple circuit we know
sum of EMFs = sum of pds
EMF = terminal pd + lost volts
E = V + Ir

E = IR + Ir

E = I (R + r)

The equations in the box to the right of the circuit will also be useful. Often the first thing you
have to do is use one of them to calculate the current in the circuit, as I appears in all versions
of the main equation.
A worked example is on the next page.
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Electricity and Electronics
Internal Resistance
Example
r
E
V
R = 28 Ω
A cell of emf 1.5 V is connected in series with a 28Ω resistor.
A voltmeter measures the voltage across the cell as 1.4 V.
Calculate:
(a) the internal resistance of the cell
(b) the current if the cell terminals are short circuited
(c) the lost volts if the external resistance R is increased to 58 Ω.
(a)
E = Ir + IR = Ir + V
Lost volts = Ir = E - V = 1.5 - 1.4 = 0.1 V
(b)
A short circuit occurs when
r=
lost volts
0.1
=
I
1
I=
V
1.4
= 0.05 A
=
R
28
r=
0.1
=2Ω
0.05
R = 0 (no external resistance)
I
E
1.5
= 0.75 A
=
=
R + r
r
2
(c)
Lost volts = Ir
I
E
= 1.5
=
R + r
28 + 2
= 0.05 A
Lost volts = 0.05 × 2
= 0.1 V
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Electricity and Electronics
Internal Resistance
ACTIVITY 3
Title: Emf and Internal Resistance with parallel circuits
Apparatus:
6 V battery, parallel circuit board, ammeter, voltmeter to measure the terminal
potential difference (t.p.d.).
V
A
L1
L2
Instruction
L3
• Connect up the circuit above, with all lamps unscrewed
so that they are off.
• Copy the table below.
• Note down the current (I) and corresponding t.p.d. (V) values when no lamps are screwed
in, and enter them into the table.
• Repeat the above for each case as one, then two, then three lamps are screwed in and light
up.
• Using Ohm’s Law, complete the final column of the table by calculating the value of the
internal resistance of the battery.
N o. of
b u lb s lit
C u rre n t I
(A )
t.p .d . V
(V )
“ lo s t v o lts ”
(V )
In te rn a l
re s is ta n c e (Ω )
0
1
2
3
1. What is the EMF of the battery (E)?
2. Calculate the mean value of the internal resistance.
3. Calculate the approximate random uncertainty in the mean.
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Electricity and Electronics
Internal Resistance
Graphical methods for finding EMF and internal resistance
Measuring I and V
Measuring I and R
E
E
r
r
A
R
A
R
The variable resistor in the above circuit is
used to obtain several pairs of readings for
current and potential difference.
The variable resistor in the above circuit is
a decade resistance box - values of
resistance can be read from it. It is used to
obtain several pairs of readings for current
and resistance.
These readings are used to plot a graph of
V vs I. It should look like this:
These readings are used to plot a graph of
R vs 1/I. It should look like this:
V
V
R
/I
1
I
As the graph is a straight line, it must take
the form y = mx + c. y in this graph is
potential difference, and x is current. We
take the equation E = V + Ir and rearrange
it to be in the form y = mx + c.
E = V + Ir
E - Ir = V
As the graph is a straight line, it must take
the form y = mx + c. y in this graph is
resistance, and x is 1/current. We take the
equation E = I (R + r) and rearrange it to be
in the form y = mx + c.
E = I (R + r)
/I = R + r
E
E
V = -Ir + E
R =
/I - r
V = -rI + E
R = E.1/I - r
Comparing this to y = mx + c, we can see
that
Comparing this to y = mx + c, we can see
that
• EMF is the y-intercept of the graph (c).
• r is the negative of the gradient (-m).
• EMF is the gradient of the graph (m).
• r is the negative of the y-intercept (-c).
Strathaven Academy
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Electricity and Electronics
Internal Resistance
Internal Resistance
15.
In the circuit shown, r represents the internal
resistance of the cell and R represents the external
resistance of the circuit.
When S is open, the voltmeter reads 2.0 V.
When S is closed, it reads 1.6 V and the
ammeter reads 0.8 A.
(a)
(b)
(c)
(d)
16.
What is the emf of the cell?
S
What is the terminal potential difference
when S is closed?
Calculate the values of r and R.
If R was halved in value, calculate the new readings on
the ammeter and voltmeter.
r
V
R
A
The cell in the diagram has an emf of 5 V. The current through the lamp is 0.2 A and
the voltmeter reads 3 V. Calculate the internal resistance of the cell.
r
V
A
17.
A cell of emf 4 V is connected to a load resistor of 15 Ω. If 0.2 A flows round the
circuit, what must be the internal resistance of the circuit?
18.
A signal generator has an emf of 8 V and internal resistance of 4 Ω. A load resistor is
connected to its terminals and draws a current of 0.5 A. Calculate the load resistance.
19.
(a)
What will be the terminal p.d. across the cell in the circuit below?
E = 1 .5 V
lo s t v o lt s = 0 .2 V
r
R
(b)
(c)
20.
Will the current increase or decrease as R is increased?
Will the terminal p.d. then increase or decrease? Explain your answer.
A cell with emf 1.5 V and internal resistance 2 Ω is connected to a 3 Ω resistor. What
is the current?
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Electricity and Electronics
Internal Resistance
21.
A pupil is given a voltmeter and a torch battery. When he connects the voltmeter
across the terminals of the battery it registers 4.5 V, but when he connects the battery
across a 6 Ω resistor, the voltmeter reading decreases to 3.0 V.
(a) Calculate the internal resistance of the battery.
(b) What value of resistor would have to be connected across the battery to reduce the
voltage reading to 2.5 V.
22.
In the circuit shown, the cell has an emf of
6.0 V and internal resistance of 1 Ω.
When the switch is closed, the reading on the
ammeter is 2 A. What is the corresponding
reading on the voltmeter ?
23.
In order to find the internal resistance of a cell, the following sets of results were
taken.
Voltage (V)
1.02
0.94
0.85
0.78
0.69
0.60
Current (A)
0.02
0.04
0.06
0.08
0.10
0.12
(a)
(b)
(c)
(d)
24.
Draw the circuit diagram used.
Plot a graph of these results and from it determine
(i) the emf
(ii) the internal resistance of the cell.
Use the emf from part (b) to calculate the lost volts for each set of readings and
hence calculate 6 values for the internal resistance.
Calculate the mean value of internal resistance and the approximate random
uncertainty.
The voltage across a cell is varied and the corresponding current noted. The results are
shown in the table below.
Voltage (V)
5.5
5.6
5.7
5.8
5.9
Current (A)
5
4
3
2
1
Plot a graph of V against I.
(a) What is the open circuit pd?
(b) Calculate the internal resistance.
(c) Calculate the short circuit current.
(d) A lamp of resistance 1.5 Ω is connected across the terminals of this supply.
Calculate (i) the terminal p.d.
and
(ii) the power delivered to the lamp.
Strathaven Academy
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Electricity and Electronics