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transcript
Ver 1.0 © Chua Kah Hean xmphysics 1
XMLECTURE
13 CURRENT OF ELECTRICITY AND DC CIRCUITS NO DEFINITIONS. JUST PHYSICS.
13.1 Voltages .................................................................................................................................... 2
13.1.1 Electromotive Force (emf) .................................................................................................. 2
13.1.2 Potential Difference (pd) ..................................................................................................... 3
13.2 Current ...................................................................................................................................... 3
13.2.1 Conventional Current .......................................................................................................... 4
13.2.1 Drift Velocity ....................................................................................................................... 5
13.3 Resistance ................................................................................................................................ 6
13.3.1 Ohm’s Law ......................................................................................................................... 7
13.3.2 Resistivity ........................................................................................................................... 8
13.4 Power ....................................................................................................................................... 9
13.5 Internal Resistance and Terminal Potential Difference ............................................................ 11
13.5.1 Maximum Efficiency.......................................................................................................... 13
13.5.2 Maximum Power ............................................................................................................... 14
13.6 I-V Characteristic Graph .......................................................................................................... 16
13.6.1 Ohmic Devices ................................................................................................................. 17
13.6.2 Filament ........................................................................................................................... 18
13.6.3 NTC Thermistor ................................................................................................................ 19
13.6.4 Light-Dependent Resistor ................................................................................................. 19
13.6.5 Diode ................................................................................................................................ 20
13.7 DC Circuits .............................................................................................................................. 21
13.7.1 Resistors in Parallel and Series ........................................................................................ 22
13.7.2 Potential Divider Principle ................................................................................................. 23
13.7.3 Slide Wire Potentiometer .................................................................................................. 25
Appendix A Circuit Rules ................................................................................................................ 32
Appendix B Practice Circuits ........................................................................................................... 33
Online resources are provided at https://xmphysics.com/coedc/
Ver 1.0 © Chua Kah Hean xmphysics 2
13.1 Voltages
An electric current is a flow of charges. An electric current in an electrical circuit is a flow of charges
round and round a loop. To produce such a current, an emf source is needed.
13.1.1 Electromotive Force (emf)
In every emf or voltage source, there is a non-electrical
force Femf acting on some mobile charge carriers1, that
result in excess positive and negative charges residing at
the positive and negative terminals respectively. These
excess charges establish an electric field which is directed
from the positive terminal towards the negative terminal
inside the emf source. Once this field is established, Femf
must do work against the electrical force FE to ferry
charges across the terminals. A positive charge therefore
gains EPE when it is “lifted” from the negative to the
positive terminal (inside the emf source). The EPE gained
per unit charge is called the emf of the voltage source.
The emf of a voltage source is measured in volts2. 1 V
corresponds to 1 J C-1. For example, for a 3 V dry cell, 3 J
of chemical energy is converted into 3 J of electrical
(potential) energy for every C of charge passing through the battery.
1 E.g. in a chemical cell, it is the “chemical force” pushing on ions in the electrolyte. In a generator, it is the magnetic force pushing electrons in the armature windings. 2 Don’t be misled by its name. An emf may be associated with an electric field and forces, but it is defined as a
voltage, not a force.
emf source resistor
E
emf source
Femf
FE Femf
FE
Ver 1.0 © Chua Kah Hean xmphysics 3
13.1.2 Potential Difference (pd)
If a resistor is connected across an emf source, a current flows
through the resistor. This is because the connecting wires
(assumed to be perfect conductors) causes the same voltage
that is the emf of the source to be across the terminals of the
resistor as well. Notice that the current flows from the positive to
the negative terminal of the resistor. A positive charge therefore
loses EPE when it makes its way from the positive to the
negative terminal of the resistor. The potential difference (pd)
(measured in volts) across the resistor thus represents the EPE
lost per unit charge in the resistor.
For example, if the pd across the filament of a light bulb is 3 V, it
means that 3 J of EPE is converted into 3 J of heat and light
energy, for every C of charge passing through the filament.
13.2 Current
Current is the net flow of electrical charges. Measured in amperes (symbol A), 1 A corresponds to
1 C s−1.
Consider an isolated tungsten filament in a light bulb before it was connected to a battery. As a
conductor, the filament is buzzing with free electrons moving at very high speeds of the order of
106 m s-1. The electrons are always colliding with the massive ions of the material, bouncing off at
Artist’s impression of the path taken by an
electron in the absence of an electric field
resistor
FE
FE
E
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random directions. Since everything is random, at the end of the day, as many electrons moved
forward as backward. There is no net flow of electrons. In other words, there is no electric current.
Now let’s connect the filament to a battery so that a constant leftward electric field E is developed
inside the filament. The electrons will now experience a constant electric force, goading them
rightward as they continue to be bounced in all directions. What results is a very slow drift of the
electrons (amidst the random collisions) as a group in the rightward direction. This net flow of electrons,
constitutes an electric current.
The drift speed is typically of the order of 10-4 m s-1. So it takes hours for an electron to trudge its way
from a battery to a light bulb. Yet a light bulb lights up the moment the switch is closed. This is because
the electric field is set up throughout the circuit immediately (in fact, this signal propagates at the
speed of light). The bulb lights up the moment the electrons in the filament starts moving.
13.2.1 Conventional Current
An electric current is a flow of charges. But there are two kinds of charges: positive and negative. It
turns out that a 1 C s-1 net flow of positively charged particles in one direction produces the exact
same electromagnetic effect as a 1 C s-1 net flow of negatively charged particles in the opposite
direction. As such, scientists have chosen to define the direction of current to be the direction of flow
of positive charges. This came to known as the conventional current.
E
Artist’s impression of the path taken by an
electron in the presence of an electric field
conventional current conventional current
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In metals, the current is carried by electrons. So while we describe the electrical current to be from
the positive terminal of the battery to the negative, the electrons in the copper wire actually flow from
the negative terminal of the battery to the positive terminal.
It is not uncommon to have currents formed by moving positive charges. For example, in an ionized
gas (plasma), or in an electrolyte, the current is carried by positive and/or negative ions. In a
semiconductor (e.g. silicon, germanium), the current is carried by valence holes (+ve) and/or
conduction electrons (−ve).
13.2.1 Drift Velocity
Consider a cylindrical wire in which electrons drifting at speed v produce a current I. What is the
relationship between v and I?
In time duration t, each free electron in the wire would have drifted forward a distance of vt. This
means that the charge carriers in the shaded volume would have passed through cross sectional area
A of the wire.
This cylindrical space has a volume of
( )( )V A v t
If there are n electrons per unit volume, the number of electrons in this volume is
N nAv t
Since each electron carries a charge of 191.60 10 Ce , the amount of charge in that volume is
Q nAv t e .
vt
A v
I
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Since current is charge per unit time, the current is
I Q t
nAv te t
nAve
In general, the current could be carried by charged particles other than electrons. So
I nAvq
where q is the charge of each mobile charge carrier.
One insight we can gain from this formula is this. For the same current, the drift velocity is higher for
an insulator than a conductor, since an insulator has much lower mobile charge concentration than a
conductor. This means a stronger electric field and thus potential difference is required to produce
the same current. It is harder to push a current through an insulator because the fewer charge carriers
have to be pushed harder. This is the primary reason for the higher resistance in insulators.
13.3 Resistance
An electric current in a tungsten filament results from the drift of electrons. We can understand the
electrons’ drift as a result of the acceleration caused by the electric field in the filament. From the
energy perspective, we can say that electrons are losing EPE in exchange for KE.
As they drift through the filament, however, the electrons collide with the massive ions of the tungsten
metal, and lose their newly gained KE to the vibrational energy of the ions. The tungsten becomes
hot and eventually glows. This is how the EPE of the electrons is ultimately converted into heat and
higher
EPE
lower
EPE
Artist’s impression of electrons passing
their KE to the metal ions
Ver 1.0 © Chua Kah Hean xmphysics 7
light in the filament. Anyway, the way electrons keep losing their energy represents a resistance to
the flow of electric current. The filament is an electrical resistor.
13.3.1 Ohm’s Law
So to maintain a current I through a resistor, a pd across the resistor V is required. It is known that for
some materials, especially metals, at a given temperature, V is proportional to I.
V IR
This relationship is called Ohm’s “Law”.
Ohm’s Law is one of those laws that are not necessarily obeyed all the time (similar to Hooke’s Law
and the Ideal Gas equation). But whether Ohm’s Law is obeyed or not, the concept of resistance is
now defined. The resistance between the points in a circuit is simply the ratio of the pd (between the
two points) to the current (through the two points).
Thus, given any mysterious two-terminal component, one can determine its resistance by applying a
known voltage V across it, and measuring the resulting current I through it. The resistance is simply
the V to I ratio.
VR
I
A word of caution: In the previous chapter, the symbol V was used to denote electric potential, and
V for difference in potential between two positions or change in potential. In this chapter, the same
symbol V is often used to denote both a voltage or potential difference, and electric potential at a point.
Electricians are not very cautious with their nomenclature to say the least.
R
higher
potential
I
lower
potential
V
I
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13.3.2 Resistivity
Consider a cylindrical carbon wire of length L and cross-sectional area A. When a pd of V is
maintained across its two ends, a current I passes through the wire. The resistance of the wire is
VR
I .
What if L is doubled? To maintain the same current as before, the pd must be doubled so that the
electric field in the resistor is maintained (V
Ex
). So the new resistance is 2
' 2V
R RI
. So we
conclude R L .
What if A is doubled? For the same V and electric field strength, the drift velocity of the electrons is
unchanged. However, having double the volume means that there are now twice as many charges
doing the drift. So the new resistance is 1
'2 2
VR R
I . So we conclude
1R
A .
Note that extending L is like connecting resistors in series. So the resistance goes up. Whereas
broadening A is like connecting resistors in parallel. The resistance goes down.
Anyway, since L
RA
, the resistance of a conductor can be written as
LR
A
The constant of proportionality is called the resistivity. It is a property of the material. For example,
copper has a resistivity of 81.7 10 Ωm (at 20°C) compared to 53.5 10 Ωm for carbon. So
obviously copper wires have much lower resistance than carbon wires of the same dimensions.
Resistivity of a material is related to the mobile carrier concentration and the atomic structure of the
material. Resistivity is affected by temperature.
In a nutshell, the resistance of a component depends both on the material it is made of and its
dimensions. A short, fat, cold copper wire is a better conductor than a long, skinny, hot tungsten wire.
L
A I
V
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Material Category Resistivity at 20°C / Ω.m
Copper Conductor 81.7 10
Carbon (graphite) Conductor 53.5 10
Water Conductor 2 to 200
Silicon Semiconductor 26.4 10
Skin (dry) Insulator 43 10
Glass Insulator 1010 to 1014
Tabulated above are the resistivity of some common materials. By the way, it is the large disparity in
resistivity between conductors and insulators that makes it is so easy to confine electric currents in
electrical circuits.
13.4 Power
From the energy perspective, the emf source converts energy from non-electrical to electrical form.
In the external circuit, the resistors convert energy from electrical to non-electrical form. For example,
in a dry-cell, chemical forces “pump” the electrons to higher EPE. In the tungsten filament, light and
heat is produced when the electrons lose their EPE (during collisions with the metal ions).
Since current is rate of flow of charges Q
It
and pd across the filament is EPE converted per unit charge W
VQ
It follows that the rate of energy conversion in the filament is IV
higher
potential
lower
potential
electrical to
non-electrical
non-electrical
to electrical
I I
V
Ver 1.0 © Chua Kah Hean xmphysics 10
From Ohm’s Law, the formula for power dissipation can be expressed in three forms.
2
2
P VI
P I R
VP
R
watch video at xmphysics.com
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13.5 Internal Resistance and Terminal Potential Difference
An external load of resistance R is powered by a battery of emf E.
Since R is connected directly across the battery, the potential difference across the resistor is always
be equal to E, regardless of the resistance value of the resistor. But a zero resistance battery is
mythical beast, like the frictionless plane and the massless pulley.
In practice, all emf sources carry some inherent internal resistance. For example, a chemical cell must
push ions through the electrolyte. An electric generator must push the current through the coil
windings with thousands of turns. A solar cell must push electrons and holes through the silicon
substrate. If the resistance of the electrical path in the emf source is substantial, then a significant
potential difference is required to push the current through the emf source itself. In casual speak, we
say that part of the emf of the battery E is used up by the pd across the internal resistance r, lowering
the terminal potential difference Vt that is ultimately available to the external resistance R. This
sentence is encapsulated by the equation below
tV E Ir
R
E I
R
r E I
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Plotted as a graph, we get a downward sloping straight line whose steepness corresponds to the
internal resistance r. So the terminal pd is equal to the emf only if zero current is drawn from the
battery. This happens if the circuit is left open. So the emf is also called the open circuit terminal pd.
The larger the current drawn (when the external resistance is decreased), the lower the terminal pd.
Another expression for Vt is as follows:
t
RV E
R r
This expression comes from the potential divider principle since the emf is divided between the
internal and external resistances. It is thus clear that the terminal pd is substantially lower than the
emf of the battery when the internal resistance is comparable to the external resistance.
watch video at xmphysics.com
Vt
E
I
Ir
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13.5.1 Maximum Efficiency
Suppose we are now stuck with a battery with an internal resistance r. We are interested in how the
efficiency of our circuit is affected by the external resistance R.
The efficiency of the circuit can be written as
2
2 2
power dissipated in
power dissipated in and
R I R R
R r R rI R I r
So clearly, the larger the external resistance, the higher the efficiency. As the internal resistance
becomes a smaller fraction of the total resistance, the power wasted in the internal resistance also
becomes a smaller fraction of the total power. Note that when R r , the efficiency is exactly 50%!
100%
efficiency,
external resistance, R
50%
r
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13.5.2 Maximum Power
But what about the output power Pout? By Pout, we mean the power dissipated in the external
resistance. So
( )( )
out tP V I
R EE
R r R r
Perhaps you feel that having a large external resistance R would result in a large Pout. If R is too large,
Vt approaches open circuit voltage of E but I approaches zero. Pout approaches zero!
Perhaps you changed your mind. We should go for the smallest external resistance instead. But if the
external resistance is too small, I approaches short circuit current of E
r but Vt approaches zero. Pout
approaches zero again!
So what is the value of R that maximizes Pout? Let return to the math.
2
2( )out
RP E
R r
By differentiating the equation and solving for 0dP
dR or otherwise, it can be shown that maximum
Pout is achieved when the external resistance R matches the internal resistance r.
2
max 2
2
( )
4
rP E
r r
E
r
Pout
external resistance, R r
E2/4R
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In theory, if your battery has zero internal resistance, infinite output power is attained by an external
resistance of zero. In practice, however, maximum power of 2
4
E
r is attained by an external resistance
that matches the internal resistance of the battery. This is called the maximum power theorem.
Recall that when R r , the efficiency is only 50% (because half the power is wasted in the internal
resistance). That’s the price to pay if high power output is what you’re after. For example, if you want
your headphone to be playing at the loudest possible volume, the electrical resistance of your
headphone should be designed to match the internal resistance of the circuitry driving your
headphone.
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13.6 I-V Characteristic Graph
Suppose you are given this mysterious two-terminal component or device. To investigate its behavior,
you can apply different voltages across its terminals, and measure the resulting currents passing
through it. If you plot the data as a current-voltage graph, you have produced the I-V characteristic
curve for this component.
Suppose the I-V curve for this (fictitious) component looks like this. Can you tell from the curve the
resistance of this component?
Remember that resistance is a simple voltage to current ratio. So the calculation of the resistance of
the component at different operating points is straight forward.
3.06.0 Ω
0.5AR
6.01.6 Ω
3.8BR
9.01.9 Ω
4.7CR
So, clearly this component’s resistance varies from one operating point to another. The resistance at
each operating point is actually equal to the reciprocal of the gradient of the line joining the origin and
that operating point. I like to call these the wiper lines (drawn in blue in the diagram).
The lower the wiper line leans towards the V-axis, the larger the resistance. The higher the wiper line
leans towards the I-axis, the smaller the resistance. Using this idea, even without calculation, we
could have figured out that B C AR R R .
I/A
V/V A (3.0, 0.5)
B (6.0, 3.8) C (9.0, 4.7)
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13.6.1 Ohmic Devices
The resistance of a coil of copper wire is constant as long as its dimensions and temperature are kept
constant. Likewise, a carbon film resistor presents a fixed resistance when operated at low power.
Resistors which obey Ohm’s Law (V IR ) are said to be ohmic. Obviously, the I-V graphs for ohmic
resistors are straight lines passing through the origin. So the lines themselves are the “wiper lines”.
The lower the line, the higher the resistance. So 1 2 3R R R .
Most off-the-shelve resistors can be assumed to be ohmic as long as they are operated at low power.
When operated at high power, their temperature may change significantly and their resistances may
be affected.
I/A
V/V
R1
R2
R3
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13.6.2 Filament
The filament of an incandescent bulb is typically made of tungsten. Like most metals, the resistivity of
tungsten is highly dependent on its temperature. At higher temperature, the metal lattice vibrates more
vigorously and saps more energy from the passing electrons. It corresponds to a higher resistivity,
resulting in an increase in the resistance of the filament.
As a result, the I-V graph of a filament is a flattening curve at high V values. From the fact that the
“wiper lines” lean more and more towards the V-axis, we can tell that the resistance of the filament
increases with V. This is because as the voltage across the filament increases, so does the current,
and thus the power dissipation ( )P VI . Higher power dissipation results in a hotter filament, which
has a higher resistance. This is why the I-V graph of a filament is straight only at low voltage and
power. Once the voltage and power is high enough and the filament starts to get hot, the graph starts
to flatten.
watch video at xmphysics.com
I/A
V/V
A
B C
RA<RB<RC
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13.6.3 NTC Thermistor
A thermistor is a piece of semiconductor. A semiconductor
has very interesting properties. At very low temperatures, a
semiconductor is basically an insulator because it does not
have any mobile charge carriers to carry a current. At higher
temperature, electrons are “shaken loose” from the atoms
and become mobile. So the higher the temperature, the
higher the charge carrier concentrations (more charge
carriers per unit volume). Hence the resistivity of a
semiconductor decreases as temperature rises. Hence the
resistance of a thermistor decreases with temperature.
As the I-V graph reveals, a thermistor has a different constant resistance at different temperatures.
The higher the temperature, the lower the resistance.
13.6.4 Light-Dependent Resistor
An LDR is a piece of semiconductor designed to be exposed
to external illumination. At very low temperatures, a
semiconductor is basically an insulator because it does not
have any mobile charge carriers to carry a current. The
illumination provides the energy to “loosen” some electrons
from the atoms. These electrons are mobile and capable of
carrying a current. So the more brightly the LDR is
illuminated, the higher the charge carrier concentrations
(more charge carriers per unit volume). Hence the resistivity
of the semiconductor decreases with illumination. Hence the
resistance of a LDR decreases with illumination.
As the I-V graph reveals, a LDR has a different constant resistance at different intensity L of the
illumination. The more light it receives, the lower the resistance.
I/A
V/V
T1
T2
T3 T1<T
2<T
3
I/A
V/V
L1
L2
L3 L1<L
2<L
3
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13.6.5 Diode
A diode aka p-n junction is another interesting semiconductor device. It is a two-terminal device: one
terminal is called the p junction and the other terminal is called the n junction. When you connect a
diode to an electrical circuit, you better know which terminal is which because the diode allows the
current to flow in one direction only.
If the voltage applied is such that the p junction is at a higher potential than n junction, the diode is
said to be in forward bias. Conversely, if the voltage applied is such that the n junction is at a higher
potential than the p junction, the diode is said to be in reverse bias.
An ideal diode should present zero
resistance in forward bias, and infinite
resistance in reverse bias. Basically, an
ideal diode is a perfect one-way valve.
A practical diode requires a small amount of
forward bias (called the turn-on voltage)
before it switches on. Nevertheless, with
sufficient forward bias, it resistance
decreases rapidly and soon behaves like a
0- wire connection. A practical diode also
allows a very tiny amount of current (called
the reverse current) in reverse bias.
Nevertheless, the resistance remains large
and it behaves practically like an open circuit.
p junction n junction
forward
bias
+ − I
reverse
bias
+ − I=0
short
circuit
open
circuit
V/V
I/A
V/V
I/A
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13.7 DC Circuits
Electrical and electronics engineering is basically about getting charges to do work and perform tasks
for us. From power generation to motors to antenna to computers and handphones, electrical circuits
are necessary to channel the charges and current into paths we want them to take. In the H2 syllabus,
we are really only scratching the surface, playing with just batteries and resistors, with an occasional
diode or thermistor. Really basic stuff.
Ver 1.0 © Chua Kah Hean xmphysics 22
13.7.1 Resistors in Parallel and Series
When resistors are connected in series, both resistors have the same current passing through them.
And the pd across each resistor sum up to be the total pd. So
1 2
1 2
1 2
total
eff
eff
V V V
IR IR IR
R R R
When resistors are connect in parallel, both resistors have the same pd across their terminals. The
currents through each resistor sum up to be the total current. So
1 2
1 2
1 2
1 1 1
total
eff
eff
I I I
V V V
R R R
R R R
The derivation can be easily extended to more than two resistors.
Connecting N resistors (with resistances R1, R2, … RN) in series results in an effective resistance of
1 2 ...eff NR R R R
Connecting N resistors (with resistances R1, R2, … RN) in parallel results in an effective resistance of
1 2
1 1 1 1...
eff NR R R R
R1 R2
V1
I
V2
Vtotal
Reff I
Vtotal
R1
R2
Itotal
V
I1
I2
Reff Itotal
s
V
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13.7.2 Potential Divider Principle
Two resistors R1 and R2 are connected in series. If the total voltage across them is Vtotal, how is this
voltage divided between R1 and R2?
Since the current following through them is the same, 1 1
2 2
1 2( )total
V IR
V IR
V I R R
From here, it is obvious that the resistor with the larger resistance grabs a bigger share. In fact, the
voltage ratios follow the resistance ratios. This is called the potential divider principle.
1 1 11
1 2 1 2
2 2 22
1 2 1 2
total
total
total
total
V IR RV V
V IR IR R R
V IR RV V
V IR IR R R
A potential divider is a very frequent occurrence in practical circuits. In fact, a potential divider
(sometimes called a potentiometer) is a standard off-the-shelve three-terminal component, with the
symbol
As the symbol suggests, the third terminal is connected to a sliding contact, called a wiper, moving
over the resistive element.
R1 R2
V1 V2
Vtotal
I
12 V
Vout
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For example, in the circuit above, by adjusting the sliding contact, Vout can be varied between 0 V and
12 V. Vout can then be used as a signal to control the brightness of a lamp, or the volume of the stereo.
In the circuit above, a thermistor and a fixed resistor form a potential divider.
thout in
th
RV V
R R
Since the resistance of the thermistor increases with temperature, this circuit can be used to “sense”
the temperature. For example, we can mount this thermistor on your laptop CPU so that the thermistor
can track the temperature of the CPU. This works because Rth (and thus Vout) decreases when the
CPU is hot and increases when the CPU is cool. So Vout could be used as an input signal to a
microcontroller to decide when to turn off the cooling fan.
In the circuit above, a LDR and a fixed resistor form a potential divider.
out in
ldr
RV V
R R
Since the resistance of the LDR increases with illumination, this circuit can be used to “sense” the
intensity of the ambient lighting. For example, we can mount this LDR on the roof top. So Vout would
increase when it is bright as Rldr decreases, and increases when it is dim as Rldr increase. So Vout
could be used as an input signal to a microcontroller to decide when to turn off the lights.
R
Rth
o
o
Vout
Vin
R
Rldr
o
o
Vout
Vin
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13.7.3 Slide Wire Potentiometer
A special application of the potential divider is the slide wire potentiometer circuit. It is basically an
elaborate “ruler” for measuring voltages. Since we have voltmeters and DMMs (digital multimeters)
nowadays, I don’t think anybody uses such a circuit anymore. Nevertheless it serves as a decent
introduction to the concept of potential balancing.
The easiest way to understand how this “voltmeter” works is to use an example.
In this example circuit, we have cell of known emf 1 6.0 VE connected across a long thin wire of
uniform cross sectional area, length 60 cmL and total resistance 6.0 ΩR . This is called the
driver circuit of the potentiometer. This cell is often called the driver cell. The wire is called the slide
wire.
Do you notice the other cell of emf 2 3.0 VE ? One end of the cell is connected to the positive end
of the slide wire. The other end is connected to a galvanometer and a jockey. The jockey can be made
to contact any point along the slide wire. This is called the secondary circuit. In practice, E2 is the
unknown emf we are trying to measure. I am revealing its value to you to make the explanation easier.
Do you realize that we are looking at a two-loop circuit driven by two emf sources? In general, solving
such a circuit involves writing two Kirchoff’s Law equations. But you don’t have to worry about that.
Thanks to null deflection!
galvanometer
A B
@ 30 cm
X
E2=3.0 V
E1=6.0 V
C D
jockey
60 cm, 6.0
Ver 1.0 © Chua Kah Hean xmphysics 26
Consider point X to be the midpoint of AB (the 30 cm mark). Before the jockey is connected, there is
no current flowing between the driver and secondary circuit. So the current in the driver circuit (and
slide wire) can be calculated easily to be 1 3 1.0 AI I . And 3.0 VAXV . On the secondary side, we
have 3.0 VCDV . So before the jockey is connected, we already have AX CDV V . Therefore when the
jockey is connected, no current will flow between the two circuits. And the galvanometer shows null
deflection, as shown below.
A B
@ 30 cm
X
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=1.0 A
I2=0.0 A
I3=1.0 A
VAX
=VCD
=3.0 V
A B
@ 30 cm
X
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=1.0 A
I2=0.0 A
I3=1.0 A
VAX
=VCD
=3.0 V
Ver 1.0 © Chua Kah Hean xmphysics 27
Now consider point Y to be at the 20 cm mark of the wire. Before the jockey is connected, the current
in the slide wire can be calculated easily to be 1 3 1.0 AI I . And 2.0 VAYV . On the secondary
side, we have 3.0 VCDV . So before the jockey is connected, we have AY CDV V . When the jockey
is subsequently connected, a current must flow from C to A to increase the current in the slide wire
so that VAY can be increased to match VCD. And the galvanometer shows a deflection. In case you’re
interested, I have presented the solution to this circuit below. You don’t have to know how I obtained
these answers. But you can check and verify that this is indeed the correct solution because (1) the
currents tally up at each junction and (2) the voltages tally up in each loop.
A B
@ 20 cm
Y
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=1.0 A
I2=0.0 A
I3=1.0 A
VAY=2.0 V, VCD=3.0 V
A B
@ 20 cm
Y
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=0.75 A
I2=0.75 A
I3=1.5 A I1=0.75 A
VAY=VCD=3.0 V
Ver 1.0 © Chua Kah Hean xmphysics 28
Now consider point Z to be at the 40 cm mark of the wire. Before the jockey is connected, the current
in the slide wire can be calculated easily to be 1 3 1.0 AI I . And 4.0 VAZV . On the secondary
side, we have 3.0 VCDV . So before the jockey is connected, we have AZ CDV V . When the jockey
is connected, a current must flow from A to C to decrease the current in the slide wire so that VAZ
decreases to match VCD. And the galvanometer shows a deflection in the other direction. In case
you’re interested, I have presented the solution to this circuit below. You don’t have to know how I
obtained these answers. But you can check and verify that this is indeed the correct solution because
(1) the currents tally up at each junction and (2) the voltages tally up in each loop.
A B
@ 40 cm
Z
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=1.0 A
I2=0.0 A
I3=1.0 A
VAZ=4.0 V, VCD=3.0 V
A B
@ 40 cm
Z
E2=3.0 V
E1=6.0 V
C D
60 cm, 6.0
I1=1.5 A
I2=0.75 A
I3=0.75 A I1=1.5 A
VAZ=VCD=3.0 V
Ver 1.0 © Chua Kah Hean xmphysics 29
So do you realize how the potentiometer works now? Basically, to measure E2, we slide the jockey
along the slide wire until we find the null deflection point. Do realize that AX CDV V all the time
regardless of where the jockey makes contact (since A is connected to C and X is connected to D).
What’s special about the null deflection point is that when there is no current flowing between the
driver and secondary circuits (zero current in DX means zero current in AC also), the two-loop circuits
are no longer intertwined with each other. This allows us to analyze the two circuits separately as if
they are not connected. So based on the ratio of the balance length LAX to the total length LAB of the
slide wire, we have
AXAX AB
AB
LV V
L
For this simple example, we also have
2AX CDV V E
1ABV E .
So the emf of the secondary cell is simply 2 1
AX
AB
LE E
L .
Primary circuit: 1
AXAX
AB
LV E
L
Secondary circuit: 2CDV E
The connection: AX CDV V
A B
@ LAX
X
E2
E1
C D
LAB, R
I1
I2=0.0 A
I3=I1 I1
VAX=VCD
Ver 1.0 © Chua Kah Hean xmphysics 30
In practice, the circuit can be slightly more complicated. For example, a protective resistor R2 may be
added to the secondary circuit. Without this resistor, a large current may be flowing in the secondary
circuit as we are still searching for the null deflection (by tapping the jockey too close to A). However,
having R2 does not affect our calculation of E2 at all. This is because at null deflection, there is no
current flowing through R2 so the pd across R2 is zero. So VCD is equal to E2 at null deflection, whether
R2 is there or not.
We also often include a resistor R1 in the driver circuit. This would make VAB a fraction of E1. Usually
R1 is added so that the balance length LAX is increased. This is to reduce the percentage uncertainty
in the measurement of LAX and ultimately of E2.
With R1, E2 is obtained through the following equations:
Primary circuit: AXAX AB
AB
LV V
L and
1
1
AB
RV E
R R
Secondary circuit: 2CDV E
The connection: AX CDV V
A B
@ LAX
X
E2
E1
C D
LAB, R
I1
I2=0.0 A
I3=I1 I1
VAX=VCD
R1
R2
Ver 1.0 © Chua Kah Hean xmphysics 31
The circuit shown above is one where the potentiometer is used to measure the terminal pd of E2. It
looks complicated. But the null deflection saves the day by allowing us to analyze the driver and
secondary circuits separately.
Primary circuit: AXAX AB
AB
LV V
L and
1
1
AB
RV E
R R
Secondary circuit: 32 2 3
3
CD
RV E E I r
R r
The connection: AX CDV V
A B
@ LAX
X
E2
E1
C D
LAB, R
I1
I2=0.0 A
I3=I1 I1
VAX=VCD
R1
r
R3
I3
Ver 1.0 © Chua Kah Hean xmphysics 32
Appendix A Circuit Rules
Some circuits are actually quite fun. As long as you are clear of the “rules”, and as long as you have
enough practice.
(1)
The pd across a zero-ohm wire must be zero. Because the tiniest voltage would result in an infinitely
large current. 0
V VI
R . Conclusion: any points connected by a wire are always at the same
electric potential.
Points of the same potential are like points on the same floor in a building.
(2)
A resistor (with non zero resistance) must have a pd across it when a current runs through it. The
current enters the resistor from the higher potential terminal, and exits at the lower potential terminal.
On the other hand, if no current is running through the resistor, then its pd must be zero, and both
ends of the resistor are at the same potential. .0 0V IR I
Resistors are like staircases which may connect a higher floor to a lower floor.
(3)
An emf source is like an elevator that brings charges from ground floor to top floor. Resistors are like
staircases that charges take to make their way back to ground floor. This is why the total emf must
be equal to the total pd in the circuit.
(4)
For a current to flow, a mobile charge must move into the space vacated by another mobile charge.
If the mobile charges at any point in the circuit do not budge, then the flow of charges grinds to a
complete half throughout the circuit. Basically everyone must move together, if not, nobody can move
at all. This is why there is zero current if the circuit is open. This is also why the total current entering
any node is equal to the total current leaving.
(4)
Two branches are parallel if they start at the (potential) point, and end at the same (potential) point.
The pd of the branches are the same.
Ver 1.0 © Chua Kah Hean xmphysics 33
Parallel branches are like two staircases that start from the same higher floor, and end at the same
lower floor.
Appendix B Practice Circuits
1)
What is the resistance between P and Q?
2)
Evaluate the resistance between i) BC and ii) AC.
P Q R R R
20
50
30
20
50
B
A
C
D
Ver 1.0 © Chua Kah Hean xmphysics 34
3)
2016 P1 Q29
A battery of emf 24 V and negligible internal resistance is connected to a network of resistors.
What is the potential difference between junctions X and Y?
4)
What do the voltmeters read if bulb B is fused?
5)
What is the current in the 2 resistor?
20 V
X
6.0 Ω
4.0 Ω
4.0 Ω 8.0 Ω 8.0 Ω
Y
4.0 Ω
9 V
A
B
4
4
2
12 V
12 V
Ver 1.0 © Chua Kah Hean xmphysics 35
6)
Above are two circuits for determining the unknown resistance R. An ideal ammeter has zero
resistance. An ideal voltmeter has infinite resistance. For most resistance R, both circuits are equally
good. However, if resistance of R is so small that it is comparable to the ammeter’s resistance, then
only one circuit is good. Conversely, if R is so large that it is comparable to the voltmeter’s resistance,
then the other circuit is good. Which is which?
7)
For both circuits, determine the range of voltage across the fixed 10 resistor.
R
E
I
I
R
E
I
I
(X) (Y)
0 to 100
10
9.0 V
10
9.0 V 100
Ver 1.0 © Chua Kah Hean xmphysics 36
8)
Evaluate the resistance between i) BC and ii) AD.
Answers:
1) R/3
2) 5.77 Ω, 13.7 Ω
3) 4.0 V
4) 0 V across A, 9 V across B
5) 0 A
6) Use X for very small R, Y for very large R
7) 0.82 V to 9.0 V, 0 V to 9.0 V
8) R/2, R
R
B
A
C
D
R
R R
R