CHAPTER 2
ELECTRICAL SYSTEMS AND PROCESSES
In this chapter, we introduce electric phenomena and the ideas and concepts needed
to describe simple electric systems and processes. There is much more to electricity
(and magnetism) than we can discuss here. We will concentrate on aspects that are
similar to the storage and flow of fluids. For example, it is quite customary to produce electrical models to represent blood circulatory systems.
Second, we go further in using analogies and introduce some simple elements of
gravitational processes. Fluids, gravity, and electricity will be found to be coupled
by the energy principle which is discussed in a separate chapter. Here we assume
that readers are familiar with the most basic aspects of the role of energy in physical
processes, and can apply the ideas to electric, gravitational, and fluid processes.
2.1 SOME I MPORTANT OBSERVATIONS
Electric phenomena are observed in everyday non-technical situations when certain
materials are rubbed. One such material is amber, whose Greek name is elektron
which gave electricity its name. This is probably how electricity was discovered.
Machines were built some three hundred years ago that could amplify the effect of
what we call static electricity. It was found that lightning could produce some of the
same effects, showing that the atmosphere is a giant electric generator. Galvani and
Volta discovered how electricity could be made to flow in a sustained manner with
the help of galvanic cells and volta piles—forerunners of today’s batteries. Some
200 years ago, this made the scientific and technical study of electricity possible.
Charging and discharging spheres. A way to demonstrate electric effects resulting
from rubbing materials is the following. A rubber stick is rubbed with some animal
fur and a metal sphere mounted on an insulating stand is touched with the rod. We
can sometimes feel the presence of electricity directly with our bodies, or we can
use a so-called electrometer to show that the sphere has been charged (Fig. 2.1).
Figure 2.1: A metal sphere is electrically charged
with a rubber rod that was rubbed with some animal fur. The electrometer (right) shows the presence
of electricity on the sphere.
An electrometer is made of two metal parts one of which is movable. They are sus-
PHYSICS AS A SYSTEMS SCIENCE
21
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
pended from an insulated mount. When they are electrically charged, they move
apart (Fig. 2.1, right), showing the presence of electricity. When an electrometer is
attached to a sphere by a metal wire it serves as a measuring device.
Note that you can get a reading of zero again very simply by touching the sphere
with your hand. Obviously, the charged sphere can be discharged through your
body. If the sphere is strongly charged, you may feel a faint shock, and quite possibly you might notice a spark flying from the sphere to your finger (you can hear the
discharge accompanying the shock). This spark is easily observed if you charge a
sphere and then bring a second uncharged sphere close to it without touching it.
A related phenomenon can be demonstrated as follows. Use two spheres of the type
described above. Charge one with the help of the rod rubbed with fur. Now use a
small metal ball or plate attached to an insulating handle. Touch the charged sphere
with the small metal piece while holding the insulated handle, move your tool from
the first sphere to the uncharged one and observe that the reading of the electrometer
of the second sphere increases. Then repeat the procedure. Each time, the reading
increases further, albeit a little less every time until there is no change any longer.
Interpretation. The last phenomenon suggests the image of “spooning some stuff”
from the charged sphere to the uncharged one. Each time the quantity of this stuff
increases on the second sphere as indicated by the electrometer. Since the reading
moves up less and less, it appears that spooning becomes more difficult. This is
rather similar to moving water from a tank into another through a connecting hose.
The electricity placed on (or in) a sphere can be made to disappear easily and quickly by touching the sphere, or by connecting the sphere with the ground with the help
of a metal wire. On the other hand, the sphere remains charged for a long time if it
is mounted with the help of some materials such as rubber or plexiglass. This can
be interpreted as follows. The human body is a fairly good conductor, a metal wire
is a very good one, and rubber and plexiglass are electric insulators. Electricity can
also flow off through the air, generating a spark.
The electric stuff which charges a sphere, which can be “spooned” and which flows
off through a wire or through the human body is called electric charge.
Establishing electrical equilibrium. Use two spheres each with an electrometer and
charge one of them. If you connect the spheres with a glow lamp, the lamp will glow
for a brief moment, and the readings of the electrometers become equal (Fig. 2.2).
5
V
V
UC1
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Time / s
30
40
Figure 2.2: Left: Differently charged bodies in (electrical) contact: Charge flows from the one more strongly charged. Flowing
charge can light a glow lamp. The phenomenon can be observed under controlled conditions with two connected capacitors
(second from left; the parallel lines in the circuit diagram are the symbol for capacitors, a rectangle is a symbol for a resistor, V
stands for volt-meter, U for voltage). The diagram on the right shows the voltages of the capacitors as functions of time.
22
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
There is a technically relevant equivalent of this phenomenon which allows measurements to be made quite easily. Build an electric circuit of two so-called capacitors (they are the equivalent of the two spheres). Charge one of the capacitors with
the help of a battery or a power supply and connect both by a conductor (a so-called
resistive element). Use volt-meters to measure the voltage across each of the capacitors as a function of time. The results is seen in the diagram on the right of Fig. 2.2.
The voltage of the charged capacitor decreases, and the reading for the uncharged
one increases. The readings change until they have become equal.
Interpretation. Note the strong similarity of the curves in the graph of Fig. 2.2 and
those measured when letting a liquid flow from one tank into another container
through a connecting hose (Fig. 1.1). The interpretation of the hydraulic phenomenon requires two concepts, those of quantity of liquid and of fluid level or fluid pressure. We say that the liquid flows from the tank with the higher fluid level to the
one with the lower level until the levels (or pressures) have equilibrated.
This is exactly how we look at electric phenomena. We introduce a quantity of electricity—called electric charge—which can be stored in systems and which can
flow. Secondly, we imagine an intensity or level of electricity, called electric potential, whose difference is responsible for flows of electricity. The difference of electric potential is called voltage. That’s what we measure with voltmeters. As long as
there is a voltage between the two capacitors in the experiment, charge flows and
the voltages across the capacitors change until they are equal. Then the process
stops.
Since a battery can be used to supply charge to a capacitor, batteries appear to function as pumps of electric charge.
Separating charge to produce positive and negative charge. Where does the electricity come from when we rub materials such as a rubber rod against fur or a glass
rod against silk? The following phenomena allow an interesting interpretation.
A metal sphere can be charged if we pull a sticky tape off the surface of the sphere
(Fig. 2.3). The electrometer demonstrates the presence of charge on the sphere. If
we touch a different sphere connected to a second electrometer with the sticky tape,
the electrometer shows charge on the tape. If we then bring a rubber rod rubbed with
fur close to the electrometers, we notice that the reading on one of the meters increases whereas on the other one it decreases. Moreover, if we let the two spheres
touch, the readings of both electrometers go back to zero.
Interpretation. Charge is not produces when materials rub against each other. Rather, positive and negative charge is separated by the process. When we bring two
bodies carrying equal amounts of positive and negative charge in contact, the two
amounts neutralize each other—we have again zero charge. When we bring a
charged rubber rod (or any other charged insulator such as glass or amber) close to
oppositely charged electrometers (or touch the electrometers), the reading of the
meter carrying charge of the same sign as the rod will increase (addition of charge
of equal sign) whereas the reading of the other meter will decrease (addition of
charge of opposite charge or neutralization).
Today we believe that electric charge can neither be created nor destroyed. When
we “produce electricity,” positive and negative charge is separated. When electricity “disappears,” positive and negative charge neutralize each other.
Positive and negative signs of charge have to be defined once. After that, signs can
PHYSICS AS A SYSTEMS SCIENCE
23
NEUTRAL
+ + +
+ +
––––
––––
Figure 2.3: Sticky tape attached to a metal sphere
on an insulating stand (top left). When the tape is
pulled off, the attached electrometer shows the
presence of charge on the sphere. It turns out that
the tape carries an equal amount of electric
charge, but of opposite sign. We say that positive
and negative charge have been separated out of
a neutral state.
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
be established by comparison. Amber rods carry negative charge.
Circuits: A battery, wires, a light bulb, and ammeters. When experiments are performed with static electricity as described above, electric processes are often weak
and cannot be sustained for long time. The discovery of the galvanic effect by the
medical doctor Luigi Galvani (1737 – 1798) and the subsequent development of
volta piles (combinations of galvanic cells) by Alessandro Volta (1745 – 1827)
changed all that. Today’s batteries which are based on Volta’s development can be
used to demonstrate important electric phenomena.
Ammeter
Flow of charge
Flow
A
Hose
Pump
+
Turbine
Battery or
power supply
Lamp
A
Hose
Flow
Ammeter
Figure 2.4: A lamp is hooked up to a battery (or to a power supply). To do so we need two wires, one going from the positive terminal
of the battery to the lamp, the other going from the lamp to the negative terminal of the battery. Ammeters which measure currents
of charge can be placed in the circuit. Center: Electric circuit diagram. Right: Analogous hydraulic circuit.
Junction
IQ1
+
IQ2
Lamp
IQ3
Lamp
Figure 2.5: A branching circuit with two loops. A
charge current flows toward a junction and splits
into two parts which add up to the original current.
A small incandescent bulb (or a small motor) is connected to a battery. This is done
with two wires. The first goes from the positive terminal of the battery to the lamp,
the second goes from the lamp to the negative terminal (Fig. 2.4). It is important
that the electric system has a closed circuit. If it is open at a point, if one of the wires
is not connected, the lamp will not burn.
We can place one or two ammeters into the circuit. Basically, a wire could be cut in
half, and the ends can be connected to the meter. It is observed that two meters, one
in the upper part of the circuit in Fig. 2.4 (center) and one in the lower part, show
the same readings. The readings are zero if the circuit is not closed.
Interpretation. Batteries or power supplies serve as pumps for electric charge. Here
they pump charge through an electric circuit made of the battery, wires, and a lamp.
The ammeters measure the flow of electricity, i.e., they quantify charge currents.
Since the charge currents are equal in the two branches of the circuit, the charge
flowing away from the battery returns to the battery (since there is positive and negative charge, currents of positive charge are those coming from the positive terminal
of the battery). The lamp does not use up electricity, just as little as a turbine uses
up water. Clearly, in interpreting electric systems and processes, we can make use
of a strong analogy with hydraulic circuits and processes (Fig. 2.4, right).
Circuits: Branching currents. A second lamp (or resistive element, motor, electric
pump) is added in parallel to the circuit of Fig. 2.4. This leads to a second loop and
creates two junctions in the circuit. Junctions are points where wires split or merge.
Now there are three separate branches to the circuit (four if we include the wire returning to the battery or power supply, Fig. 2.5). Measurement of the charge currents shows that the currents split at a junction. The sum of the two currents flowing
away from the junction equals the current flowing toward the junction.
24
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
Interpretation. The observation that charge currents split or merge at junctions
lends still more support to the notion that electric charge is neither produced nor destroyed. Here, it simply flows in closed circuits.
Voltages in a closed circuit. Create a circuit made of a battery or power supply and
two lamps (or other resistive elements, or electric motors). The devices will be connected in series forming a single electric circuit (Fig. 2.6, left). Use three volt
meters to measure the potential difference across each of the three elements.
Voltmeter
V
D
E
+
U1
UB
V
A
U2
V
F
Figure 2.6: Voltages across the elements in a single
loop circuit (the elements are connected in series).
The voltmeters are connected in parallel to an element. Voltages (potential differences) are denoted
by (blue) arrows parallel to the devices. The rectangles in the circuit diagram symbolize resistors.
Potential
C
B
A B C D E F A
Position
It is observed that the sum of the voltages (potential differences) across the lamps
(or the other devices) equals the voltage across the battery (or power supply). Alternatively, if we take into consideration the signs of the voltages, we see that the
sum of the three voltages equals zero.
Interpretation. Previously, we have interpreted voltages as electric potential differences and the electric potential as a kind of electric level comparable to fluid pressure in hydraulics. We can continue to make use of the analogy between electricity
and hydraulics and view the electric potential along a circuit as a level in a landscape (Fig. 2.6, right). Going around a circuit we go up and down in this landscape.
When we are back at the starting point, we have gone up as far as we have come
down. The level differences (voltages) must add up to zero, just as pressure differences in a closed hydraulic circuit add up to zero (Chapter 1).
If we orient ourselves according to the direction of flow of electric charge in the
simple circuit of Fig. 2.6, the potential goes up when we go through the battery from
A to B. In resistive elements or motors, i.e., from C to D and from E to F, the potential goes down. Wires in circuits are typically modeled as ideal conductors where
the potential does not change (there is no voltage along an ideal wire).
QUESTIONS
1.
Consider the experiment described in Fig. 2.2. Do the electrometers connected to charged
spheres measure quantities of electricity (charge) or the intensity of electricity (voltage)? Why?
2.
When you rub non-conducting materials (plastic, rubber, glass, amber) on fur or cloth, static
electricity develops. Where does the charge come from? Has it been produced?
3.
A metal coated ping-pong ball is hanging from a thin thread. It can be charged with the help of
a plastic rod rubbed against fur. After this, when we bring the rod close to the ball (without
touching it), the ball is repelled. If we bring a glass rod rubbed with a silk cloth near the charged
ball, it is attracted by the rod. How can this be explained?
4.
Do generators, power supplies, and batteries generate electricity (electric charge)?
5.
Consider the circuit in Fig. 2.5. Should the charge currents through the two lamps be equal?
What relation do they satisfy in general?
PHYSICS AS A SYSTEMS SCIENCE
25
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
6.
In Fig. 2.6 (right), the electric potential in a single loop of a circuit is shown as a function of
position along the circuit. How many voltages are there? Identify them in the diagram on the
right. What is the relation between the voltages?
7.
Consider the circuit in Fig. 2.5. What must be the relation between the voltages across the two
lamps? Why?
Flow of electricity through metals. Electric charge can flow more or less easily
U
V
Figure 2.7: A wire which represents a resistive element is hooked up to a power supply. Voltage
across and current through the wire are measured.
1.2
0.15
[
2
Current / A
1
Figure 2.8: Resistive characteristic diagrams of metal conductors, in the linear regime (left) and for the
filament of a light bulb (right).
0.8
[
[
0.0
0.00
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0.4
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0.25
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0.50
0.75
Voltage / V
4
[
[
[
[
1.00
Current / A
IQ
+
through different materials. Depending upon the material, the type of charge transport is different. Metals make up one of the most important conducting materials.
We can investigate the transport of electric charge through metals most simply by
taking different wires having different lengths and thicknesses, composed of different materials. A wire is connected to a power supply whose voltage can be changed.
At the same time we measure the charge current through the wire. A wire is a resistive element, so it is represented by the symbol of a resistor in a circuit diagram (see
Fig. 2.7).
For a given wire, the electric current is measured for different voltages, and the result is displayed in a current–voltage diagram as in Fig. 2.8. Such a diagram is
called a characteristic diagram for a resistive element, the measured function is a
resistive characteristic curve (it is characteristic of the charge transport in the particular wire and of the dimensions of the wire).
0.10
0.05
[
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0.00 [
0.00
2.00
4.00
6.00
Voltage / V
8.00
When the voltage is increased, the charge current increases as well. If the changes
are not too great, the characteristic curves for metals are straight lines. We say that
we have a linear characteristic (diagram on the left in Fig. 2.8). For different wires
the curves are straight lines having different slopes. The slope is higher for shorter,
thicker wires, and it higher for some materials than for others.
If voltage and current increase more strongly for a given wire—as might be the case
for the filament of a light bulb—the current increases more slowly with increasing
voltage (diagram on the right in Fig. 2.8).
Interpretation. The voltage, i.e., the electric potential difference, is like a driving
force for the flow of charge through the wire. The higher the voltage, the stronger
the current, in analogy to what we know from the flow of fluids through pipes.
The slope of the straight characteristic lines in the diagram on the left of Fig. 2.8
indicates how easily charge flows through the given wire. The measure of how easily charge flows is called the electric conductance, again in analogy to hydraulics
(Fig. 1.7). The inverse of the conductance is the resistance. In the diagram on the
left in Fig. 2.8, characteristic 1 has the smallest resistance (highest conductance),
whereas the resistance of curve 4 is the highest.
26
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
PHYSICS AS A SYSTEMS SCIENCE
27
0.60
Current / A
The linearity of the characteristic curves suggests a similarity between the transport
mechanism for charge and that for fluids in laminar flow. Charge transport through
metals is said to satisfy Ohm’s relation. Indeed, ohmic and laminar transports have
this in common: the strength of the transport depends linearly upon how fast the
pressure or the electric potential change in the direction of flow (we use the term
gradient to describe how fast pressure or potential change in the direction of flow).
The nonlinearity of the characteristic in the diagram on the right (Fig. 2.8) might
suggest non-ohmic behavior. However, that is not the case. Charge flows through
materials of the same type for both diagrams of Fig. 2.8. Therefore, we should expect the transport mechanisms to be the same. The difference is caused by heat produced in the resistive element causing a strong change of temperature, leading to an
increase of the resistive property of the metal (the higher the temperature, the higher
the resistance). That is why the slope of the characteristic curve of the filament decreases with increasing voltage (and increasing current).
Diodes. When a diode is used instead of a metal wire in a circuit such as in Fig. 2.7,
the characteristic curve looks quite different (Fig. 2.9; the symbol of a diode is
shown in Fig. 2.10). The current is very small for small voltages. At a relatively
well defined voltage (here, around 0.7 V), the current rises rapidly and is more or
less independent of voltage. For negative values of the voltage (which would lead
to a reversal of the charge current) the current is effectively equal to zero.
Interpretation. Here, the conductive properties are clearly different from those of
ohmic transports. A diode lets charge pass only in one direction, not in the other. It
functions similarly to a valve in a hydraulic circuit.
A diode is made of so-called semiconductors containing different impurities (small
quantities of substances different from the main material). Semiconductor materials
such as silicon have conductivities that lie between those of good conductors and
good insulators (the conductivity of copper is some 10 billion times that of silicon
whose conductivity is about 10 billion times higher than that of glass). Usually, the
substrate is doped with small quantities of other substances which makes combinations of such “impure” semiconductors perfect materials for important devices (diodes, transistors, light emitting diodes, photovoltaic cells, thermoelectric devices,
or more).
In contrast to metals where there are many free charge carriers (negatively charged
electrons) related to the transport of charge, there are no free electrons in semiconductors. However, there is always a small number of pairs of positively charged
holes and negatively charged electrons that can transport charge. (Such hole-electron pairs can be produced, for example, by light absorbed by the semiconductor.
Also, the number of such pairs increases rapidly with increasing temperature.) Doping of the substrate with impurities changes the conductive properties of a sample.
If two differently doped semiconductors are assembled into a single unit, a diode
may result that lets charge flow only in a single direction across the interface of the
parts (see Chapter 4 for a detailed description).
Battery characteristic. How does a battery operate in an electric circuit? We can find
out simply by using the battery with a number of different resistive elements (resistors) having different resistances (Fig. 2.11, left). It is found that the voltage across
the battery drops with increasing current. The characteristic relation is close to linear.
0.40
0.20
0.00
0.00
[
[
[
[
[
[
[
[
[
[
[
[
[[[[[[[[ [[ [[[[[[[
0.25
0.50
0.75
Voltage / V
1.00
Figure 2.9: Characterisitc diagram of a diode. If the
voltage is reversed (negative values in the diagram), the charge current is effectively equal to zero.
IQ
U
Figure 2.10: Symbol of diode in a circuit diagram.
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
4
IQ
V
UB
R
Current / A
A
+
3
[
[
2
[
[
1
IQ
Ri
UB
V
UR
+
Uo
Figure 2.12: Model of a real battery. The electric resistor symbolizes resistive processes in the battery.
The circle with the + sign now stands for the (ideal)
voltage set up by the reactions. (This is a simple and
purely electric model—an equivalent circuit—of an
electrochemical device. A much better and more
detailed understanding of the operation—including discharging—of a battery will come with a consideration of the chemical processes; see Chapter
4.)
[
[
Figure 2.11: Characterisitc diagram of a battery
(right) obtained from measurements done on a simple circuit (left). R is the symbol of a resistor.
0
0
1
2
3
Voltage / V
[[
[
[
[
4
5
The voltage measured is close to the rated value of the battery (here: 4.5 V) if there
is no current of charge through the circuit. We get this if the circuit is open (or with
a resistor having an extremely high resistance); therefore, this value is called the
open circuit voltage of a battery. For very low external resistance, the current increases and the voltage drops. The voltage will reach a value of zero if we connect
the terminals of the battery by a short, thick wire having hardly any resistance at all.
The charge current associated with this point is called short-circuit current.
Interpretation. We tend to think of a battery as a device that establishes a fixed voltage (say, 4.5 volts for a 4.5-volt battery). If that were the case, the characteristic line
would be a straight vertical line in the diagram of Fig. 2.11, at a voltage of 4.5 V.
In other words, the voltage across the battery would be independent of the current
through the battery, i.e., independent of how the battery was working. It would always be equal to the open circuit voltage.
Since this is not the case, we have to modify our understanding of a battery (see
Fig. 2.12). If we assume that the open circuit voltage is the voltage set up ideally by
the chemical reactions, we can understand the decrease of the voltage with increasing current as the result of internal “losses.” Losses are the result of resistive behavior. We know that the electric potential drops across a resistor in the direction of the
flow of charge. Therefore, the battery characteristic can be understood as resulting
from the interplay of chemical reactions and the flow of charge through an internal
resistor (the part of the circuit inside the dashed rectangle in Fig. 2.12). Since the
voltage drop across the resistor increases (linearly) with the current through the battery, we now understand the characteristic diagram (Fig. 2.11).
A real battery gets warm when operated. This agrees with the fact that resistive elements are heat producing. In this regard too, electrical resistive elements behave
just like hydraulic ones (Chapter 1).
Charging and discharging capacitors. A capacitor is made part of two (connected)
circuits that allow it to be charged and discharged (Fig. 2.13, left). There is a resistive element in each of the two branches of the circuit. The charging circuit (the
branch on the left in the circuit diagram) contains a power supply or a battery. If a
previously uncharged capacitor is charged, its voltage behaves as in the diagram on
the right of Fig. 2.13. During discharging, the voltage drops as seen in the diagram
at the center of Fig. 2.13. Note that the speed of charging and discharging is different.
Interpretation. The behavior of the capacitor in the circuits shown here is analogous
to that of a fluid tank that is charged or discharged through a pipe. In fact, the charging and discharging circuits are equivalent to what we would build in a hydraulic
28
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
system (the power supply corresponds to a pump, the resistive elements replace the
pipes; see Fig. 1.13 in Chapter 1).
6.0
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Figure 2.13: Discharging and charging of a simple capacitor. Left: Diagram of a circuit that allows for charging and subsequent
discharging (R1 = 9.8 kOhm, R2 = 108 kOhm). Center: Voltage across the capacitor as a function of time, as it discharges. Right:
Voltage across capacitor during charging.
PHYSICS AS A SYSTEMS SCIENCE
29
4
Voltage / V
If we accept these analogies, we can interpret a capacitor as a storage element for
electric charge. The larger the amount of charge stored, the higher the voltage. With
standard capacitors and ohmic resistors, we get good models of the behavior of the
circuits if we assume the relation between charge and voltage of a capacitor to be
linear—as in the case of a straight walled tank filled with oil (Chapter 1).
There are two important differences between capacitors and fluid storage tanks.
Firstly, the voltage across a capacitor can be positive or negative (Fig. 2.14). The
voltage of a capacitor having negative charge (or a lack of positive charge) is negative. Clearly, there is no negative water, so there are no negative pressures.
Secondly, a capacitor used in an electric circuit is not really charged. It consists of
two elements one of which is positively charged while the other carries the equivalent negative charge—so the total charge of a capacitor is zero. Note the use of capacitors in circuits: There are two wires connected to the device, one for inflow and
one for outflow of charge during operation. If we want a hydraulic analogue of the
charging of a capacitor, we could imagine two tanks both filled to the same level.
We define this as the zero level. Charging means filling one of the tanks at the expense of the other. Even though the quantity of liquid has not changed, there now
is a pressure difference.
The simplest type of capacitor is made of two parallel metal plates which carry opposite charge. Capacitors for electric and electronic circuits are often made of two
metal foils separated by a nonconducting sheet. The sheets are rolled up tightly in
the form of a cylinder.
Driving charge apart with a battery. A battery is placed in a circuit containing two
capacitors and a resistor (except for the battery, this is the circuit of Fig. 2.2 that lets
us demonstrate equilibration of communicating charged capacitors). The capacitors
are uncharged in the open circuit. When the circuit is closed, the voltage of the first
capacitor will become positive while the other voltage will be negative (Fig. 2.14).
Interpretation. The electric circuit behaves analogously to two fluid containers connected by a hose with a pump placed somewhere in the hose. The initial water levels
are equal. When the pump is turned on, water flows from one of the tanks into the
other, raising one of the levels at the expense of the other. Here, electric charge is
2
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-4
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20
30
Time /s
Figure 2.14: Two capacitors in a single circuit with a
battery between them. The diagram shows how the
voltage of the capacitors changes with time.
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
6
6
4
4
Voltage / V
Voltage / V
pumped from one of the (uncharged) capacitors into the other capacitor, making
charge (and voltage) of the first decrease from zero, while charge and voltage of the
second increase.
Chains and networks of capacitors and resistors. Capacitors and resistors can form
chains (single-dimensional) or even two or three dimensional networks (Fig. 2.15).
If initial voltages are not the same for all capacitors, dynamics will ensue: The voltages of all the capacitors in the network will change in the course of time.
2
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400
Time / s
600
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Time / s
Figure 2.15: A (short) chain of capacitors with resistors between them (photograph at left) forms a physical model for diffusion of electric
charge. In a first experiment, all capacitors have the same capacitance, and all but the leftmost are uncharged. The graph at the center
shows the voltages across the capacitors as functions of time. Note the similarity of behavior with that obesrved in a chain of water tanks
(Fig. 1.14). In a second experiment, the chain of capacitors is allowed to discharge across an external resistor with a resistance which is large
compared to that of the internal resistors. The capacitance of the outermost capacitor is small compared to that of the others. The diagram
on the right shows the voltage across the outermost capacitor in the chain (the one closest to the discharging resistor).
Interpretation. The example of system behavior seen in the diagram at the center of
Fig. 2.15 suggests how we can interpret the system and what is happening. The particular case is analogous to the behavior of water levels in a chain of communicating
tanks (Fig. 1.14) where only the first of the tanks is filled, and the tanks do not discharge to the environment. As a consequence, we again see capacitors as storage
elements for electric charge, and resistors are like pipes connecting them in a chain.
In the example shown in the center graph, only one capacitor has charge which is
slowly distributed among the capacitors in the chain until all voltages have become
equal (the driving forces for the flows through the resistors have become equal to
zero).
Note that in the second example (Fig. 2.15, graph on the right), the outermost capacitor with a relatively small capacitance discharges quickly at the beginning and
much more slowly later on. This is what happens in a chain of tanks where the outermost tanks is very narrow compared to the others, and the chain drains to the environment.
Supercapacitors. Classical capacitors made in standard ways of standard materials
(such as metal foils) typically have very small electric capacitances (in the range of
one millionth to one trillionth of one Farad). On the other hand, capacitors made of
new materials can have capacitances up to several tens of Farads. Such capacitors
are called supercapacitors. In the diagram in Fig. 2.16, the voltage across such a supercapacitor during discharging through a simple circuit is shown.
Interpretation. Note the strong similarity between the behavior of the supercapacitor and that of the chain of capacitors in Fig. 2.15 in the graph on the right. On the
30
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
other hand, note that the discharge curve is noticeably different from that of a standard capacitor which leads to simple exponential decay as in Fig. 2.13 (graph at the
center). Analogy suggests that in the case of a supercapacitor, charge has to diffuse
out of the material that stores charge, just like through a chain of capacitors and resistors. A standard capacitor, on the other hand, stores charge only at the surface of
a metal sheet from which the charge “drains” without delay, leading to the behavior
known from draining a single tank through a pipe at the bottom.
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Figure 2.16: Discharging of a supercapacitor in a
simple circuit containing an ohmic resistor. Note the
quick drop of voltage at the start of the process.
0
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Time /s
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Electric heating of water. Consider the process of electric heating of water. An electric heater (an immersion heater) is placed in a given quantity of water at a prescribed temperature (such as 23°C). The heater is turned on, voltage and charge
current for the heater and the temperature of the water are measured as functions of
time. Typically, voltage and electric current can be kept constant. It is observed that
the temperature of the water rises steadily, at least at the beginning (Fig. 2.17).
When the process is repeated with the same quantity of water at the same initial
temperature, but with different values of voltage and electric current, the temperature of the water rises linearly again, this time at a different rate. Measurements
show that the rate at which the temperature rises is proportional to the product of
voltage and electric current. In other words, if we were to double the voltage and
the current of charge at the same time, the temperature of the water would rise four
times as fast.
Interpretation. The phenomenon demonstrates the coupling of two processes: The
electric process drives a thermal one, i.e., the heating of the water. Since we have
the same amount of water starting under equal conditions every time, we can interpret the different rates at which the temperature rises as the result of different “efforts” of the electric process. If the temperature rises slowly, electricity is not
“working so hard,” whereas if it rises quickly, the electric system is “working hard.”
It is customary to introduce the notion of power to describe how “hard” a system is
“working.” Expressed differently, the power of the electric process measures at
what rate the electric system is driving the thermal process.
If we use this concept, the experiment shows that the power of the electric heater is
higher the faster the temperature of the water is rising. On the other hand, data
shows that the temperature rises more quickly if voltage and electric current are
higher. To be precise, the product of voltage and current of charge is proportional
to the slope of the temperature curves in Fig. 2.17.
We interpret the power of a process as the measure of how strongly one process
drives another. To use another notion from everyday life, we say that a process that
PHYSICS AS A SYSTEMS SCIENCE
31
50
Temperature / °C
Voltage / V
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Figure 2.17: Temperature of water as a function of
time for three modes of operating the immersion
heater heating 800 g of water.
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
Immersion heater
Pel
IQ
ϕ1
Pth
Heat
ϕ2
Figure 2.18: A process diagram of an immersion
heater. It shows that electric charge flows into the
heater at high potential and leaves again at low
potential. As a result, heat is produced (the circle
with dot symbolizes a source). The coupling of the
electric and thermal processes is described in terms
of the rate at which the driving electric process releases energy (power Pel) which is equal to the
power of the thermal process (Pth).
is driven (and therefore is “involuntary”) needs energy, and the energy is given to
it by the process causing it. Here, electricity is the “cause,” the source of energy,
and the phenomenon of heating is the effect, the recipient of energy. Energy is a numerical measure of how much has been caused. Power is understood as the rate at
which the driving process releases energy or, vice-versa, the rate at which the driven
process uses energy (Fig. 2.18).
The coupling of processes in systems and devices can be described vividly in socalled process diagrams (Fig. 2.18). The diagram uses symbols of the processes
themselves (arrows for flows, vertical arrows for levels, circles with a dot for sources). The coupling is described by energy released and used which are symbolized
by fat vertical arrows inside the process diagram of a system. Remember that the
electric power is related to the voltage (difference of potentials) and the electric current. The experiment shows that there is a simple relation between the level difference (left side of Fig. 2.18) and the strength of the electric current, and the power
(fat arrow Pel). The latter is given by the product of the two former quantities.
Hydroelectric power plants. Hydroelectric power plants (Fig. 2.19) demonstrate a
coupling of a different kind: Vertical water flows drive electric processes. Vertical
water flows have to do with gravity, so we say that gravity drives electricity in hydroelectric power plants. Since the heating of water gave us an idea for how to express electric power (as the product of voltage and electric current), we can use
simple measurements to relate water flow and height differences to the power of the
water fall (Table 2.1).
Figure 2.19: The Grand Dixemce dam in the Swiss
alps. Right: A schematic of the lake and the penstock.
Table 2.1: Examples of hydroelectric power plants a
Hydroelectric
power plant
Current of
Mass
Im / kg/s
Vertical fall
of water
∆h / m
Voltage and
current b
UIq / V · A
UIQ / (Im∆h)
Bavona
18,000
890
137·106
8.6
Nendaz
45,000
1014
384·106
8.4
Handeck III
12,500
445
48·106
8.6
Chatelard
16,000
814
107·106
8.2
Tiefencastel
16,700
374
50·106
8.0
a. Hydraulic power plants with artificial lakes in Switzerland.
b. Product of voltage and electric current measured for the generator.
32
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
PHYSICS AS A SYSTEMS SCIENCE
33
Power plant
Pgrav
Im
ϕG1
ϕG2
Pel
IQ
ϕ1
ϕ2
Figure 2.20: A process diagram of an ideal hydroelectric power plant shows the coupling of electricity to gravity. Pgrav and Pel denote gravitational
and electric power, respectively. ϕG is the gravitational potential, ϕ the electric potential. Im is the
symbol for the current of mass (of water).
4
Voltage / V
Interpretation. Data in Table 2.1 shows, that the product of voltage and electric current (the “output” of the plant) is proportional to the product of level difference
(height of the fall of water) and water flow (measured as the current of mass of the
water). The latter is the “input” to the system. The ratio of the former to the latter is
almost constant.
A process diagram of the hydroelectric power plant (Fig. 2.20) provides an interpretation of the coupling of electricity to gravity in the plant. Water flows in at a
high level and leaves at a lower one. Since we look at the process as a gravitational
phenomenon, we measure quantity of water by its mass, and level by the gravitational potential. Water falling down from a high to a low level releases energy
which is used by the follow up electric process (there are processes in between
forming a longer chain, but we do not consider them here).
Data of the phenomena discussed here shows that the rates at which energy is either
released or used can always be expressed by the product of a level difference (voltage in electricity or difference of gravitational potential) and a current (electric current in electricity, current of mass in gravity). A waterfall serves as a vivid visual
metaphor for the concept of the power of a process (see Chapter E on energy in
physical processes).
Table 2.1 demonstrates something important: Not all processes are equally efficient. The higher the number in the last column, the higher the electric power relative to the gravitational power. The smaller the number, the less of the energy
released by the falling water is used by the electric process. In other words, some of
the energy released is still available, and we know what it does: It drives the production in heat as a consequence of mechanical resistance (friction) and electric resistance in the system. In fact, even the most efficient of the examples in the table
is not 100% efficient (theoretically, for an ideal system, the factor in the last column
should be close to 9.81 SI-units). Therefore, the process diagram in Fig. 2.20 is for
an ideal system (one that does not produce heat).
Discharging and charging batteries. A battery is hooked up to a device such as a
small lamp, motor, or pump, and the voltage across the battery is measured as a
function of time. If the system is allowed to operate for an extended period, a decrease of the battery voltage is observed (Fig. 2.21). In the end, the battery stops
working.
If the battery is analyzed, it is found that it consists of chemicals which, during its
operation, undergo reactions. At the end of the life of a battery, all the original substances have been used up, and new chemicals have been produced (see Chapter 4
on chemical processes and more on electrochemical devices).
There are battery types that can be “recharged” with the help of an electric power
supply. During recharging, the chemical reactions observed during operation of the
battery are reversed. If recharge is complete, the original substances are recovered
in the battery, and the battery can be used once more.
Interpretation. A battery is a chemically driven electricity pump. Chemical reactions release energy which sets up an electric potential difference (a voltage) across
the battery, and drives a charge current through this potential difference (Fig. 2.22).
In fact, charge flows from lower to higher potential through the battery, i.e., it is
“pumped uphill.” The voltage that can be set up by the chemical reactions depends
upon the chemical state of the battery. That is why the voltage decreases with time
[
3 [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
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[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
0
0
10000
20000
30000
Time / s
Figure 2.21: Voltage of a battery as a function of
time during extended use. The battery was hooked
up to a resistive element. (A 4.5 V battery and a resistor having a resistance of 4.75 Ω were used.)
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
as the battery gets discharged (Fig. 2.22, left). Note that discharging does not mean
a loss of electric charge but a decrease of the store of energy as the chemical substances are transformed.
Charging Battery
Discharging Battery
Figure 2.22: Process diagram showing the discharging of a battery (left). Chemical reactions release
energy which sets up an electric potential difference. Electric charge is pumped through this potential difference and picks up the energy released
by the reactions. Right: Charging of a battery. Electric charge flows downhill and releases energy. The
original chemical reactions run in reverse.
Pel
Pchem
IQ
ϕ1
Pchem
IQ
ϕ1
ϕ2
Pel
ϕ2
Note the analogy with a water pump. In an electrically driven water pump (see below), the electric process releases energy which sets up a pressure difference. Water
is pumped through the pressure difference from low to high pressure. (There is a
difference between a battery and a pump: The battery has its own energy store
whereas the pump gets its energy from the outside from an electric power supply.)
When a battery is charged, it is not charged or “filled” with electric charge; it remains electrically neutral since the current of charge flows through the battery. As
the chemical substances return to their original state, the battery is charged with energy (Fig. 2.22, right).
Operating an electric pump with a battery. Electric devices and machines are operated with the help of batteries, power supplies, or generators. A device such as an
electric pump can be hooked up to a battery with two wires (Fig. 2.23, left).
IQ
Figure 2.23: An electric pump is hooked up to a
battery. The stronger the electric current, and the
higher the electric potential difference (voltage),
the more strongly the pump operates.
+
ϕ1
ϕ2
IW
ϕ1
UB
V
Ideal pump
Ideal battery
Pel
UP
IV
Phydr
IQ
V
ϕ2
Pump
Pchem
ϕ1
Pel
ϕ2
p2
p1
It is observed that the pump operates more strongly if either the electric current or
the electric potential difference (voltage) is increased. In fact, the product of water
current and pressure difference rises in tandem with the product of electric current
and voltage.
Interpretation. The electric current is pumped from low to high potential in the battery (Fig. 2.23, center) and flows down from high to low potential in the pump
(Fig. 2.23, right). The electric potential differences across the battery and the pump
are equal, and so is the current through either device. When electric charge flows
uphill, it picks up energy, when it flows downhill, it releases energy. The rate at
which energy is picked up by the charge in the battery equals the rate at which energy is released in the pump. If the pump operates ideally, the energy released by
the electric charge falling through the potential difference equals the energy picked
up by the water current flowing through the pump.
34
PHYSICS AS A SYSTEMS SCIENCE
2.1 SOME IMPORTANT OBSERVATIONS
150
Electric current / mA
In the single loop circuit, the electric current is the same everywhere. The fact that
the power of the electric processes in the battery and in the pump are equal, can now
be used as an explanation for the equality of the voltages UB and UP. The balance
of power can be extended to explain the observation made above in Fig. 2.6: The
sum of the voltages in a single loop must add up to zero.
Photovoltaic cells. Photovoltaic cells are thin layers of semiconductors that produce
a voltage if exposed to (sun)light. An array of just 21 small cells (having a surface
area of 15 cm2 each) was exposed to the light of the son (or simply diffuse daylight)
three times under different conditions and the array’s characteristic curves were
measured (Fig. 2.24). The characteristic of solar cells is measured by hooking up
resistors having different resistances to the cells (or to the array) and determining
voltage and electric current (the circuit is equivalent to the one in Fig. 2.11 with the
array replacing the battery).
Interpretation. Photovoltaic or solar cells are similar to diodes, so their characteristic curves are similar to those of diodes (Fig. 2.9). The difference is that light falling
on them shifts the characteristic curve of the diode in the negative direction. Flipping the diagram about the horizontal (voltage) axis leads to the graphs seen in
Fig. 2.24. As a result, the cells work as generators, and except for the form of the
curves, the characteristic diagram is similar to that of a battery (Fig. 2.11). There is
a voltage across a cell even when no electric current is flowing (in fact, this is the
maximum possible value called the open circuit voltage as in the case of a battery).
The voltage decreases if a current is allowed to flow through the cells in an electric
circuit. The higher the current, the smaller the voltage—again like for a battery—
and there is a maximum current called the short-circuit current.
The short-circuit current depends more or less linearly upon the intensity of the light
falling upon a PV cell (see Fig. 2.24). In other words, the shift of the diode characteristic depends upon the intensity of the light. This tells us that the energy supplied
by the light is responsible for the photovoltaic effect. As explained for diodes, light
absorbed by the doped semiconductor produces pairs of holes and electrons, and
this produces the open circuit voltage of the cell. Since the transport for charge is
that for semiconducting materials, the model of a PV cell is different from the simple one applicable to a battery (Fig. 2.12). Diffusion of charge with the chemicals
of a battery leads to a resistive effect similar to that in a metal which is ohmic. In a
diode it is clearly not ohmic.
100
50
0
0
Assume a type of conducting device for which the electric current doubles if the voltage across
it is doubles. What is the characteristic diagram of such a device? How do you explain the meaning of conductance or resistance of the device using the diagram?
9.
Metals are said to be ohmic conductors which are said to have linear characteristic curves: double the voltage across an ohmic conductor or resistor gives double the current through the device. The filament of a light bulb is a metal, thus an ohmic conductor. Why is the characteristic
of an incandescent lamp not linear?
10. Why does the voltage measured across the terminals of a battery decrease if the electric current
through the battery is increasing? How are voltage and current related?
11. Produce a word model that explains the discharging of a charged capacitor in a simple circuit
having a resistor. Do the same for discharging of a tank containing oil through a pipe at its bottom. Compare the explanations you use in the electric and in the hydraulic cases.
PHYSICS AS A SYSTEMS SCIENCE
35
8
Voltage / V
12
Figure 2.24: Characteristic curves of a small array
of 21 solar cells in series exposed to the sun and the
sky on three different winter days. The insolation was
400, 200, and 60 W/m2 for the three curves (from the
top).
QUESTIONS
8.
4
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
12. Why does the result of charging or discharging of a capacitor (as in Fig. 2.13) demonstrate that
the voltage of a capacitor increases with increasing charge?
13. Consider an electric water pump. What is the meaning of driving process? What is the driven
process? What does energy have to do with one process driving another? What is the meaning
of electric power? Of hydraulic power?
14. Explain the role of energy in the operation of a battery. Explain terms such as driving and driven
process, power, energy storage, etc.
15. Consider a battery driving an electric pump. What is the role of energy?
16. Which fluid quantities can be compared to which electric quantities? How are they similar? Are
there differences? Are there elements of fluid systems that are analogous to elements of electric
systems?
17. Sketch an electric circuit that is analogous to the system of two communicating fluid tanks (see
Fig. 1.1). Do the same for a system of two tanks with an additional outflow (similar to the one
shown in Fig. 1.37).
18. Sketch a hydraulic system made of tanks and pipes that shows similar behavior as that of a supercapacitor during discharging.
Figure 2.25: Magnetic effects of flowing electric
charge (top: influencing a magnet; center: electromagnet), and coupling of electric and magnetic
effects (bottom: induction of a current).
Electric, magnetic, and gravitational fields. When electrically charged bodies are
brought sufficiently close to each other, the repel or attract each other. In other
words, electric charge can cause mechanical phenomena. The bodies do not have to
be in actual contact, so the effect works at a distance. We know similar phenomena
from magnetized bodies (magnets attract or repel each other) or from gravity (bodies always attract each other, such as the Earth attracts all bodies on its surface, and
it attracts the moon and artificial satellites).
Important processes in electricity and magnetism demonstrate that these phenomena are closely related. When a compass (essentially a magnetized metal needle) is
brought close to a wire through which electric charge flows, the compass is deflected, i.e., the electric current influences the magnet (Fig. 2.25, top). We also know
that strong magnets can be built from a wire made into a coil (a solenoid, Fig. 2.25,
center) and then letting electricity flow through the wire. Finally, if a current
through a solenoid is made to change in time, a current can be induced in a solenoid
near by (Fig. 2.25, bottom). This latter phenomenon is particularly interesting, since
it shows an effect at a distance that does not involve any mechanical action (motion)
whatsoever.
Interpretation. All these phenomena, from the weight of a stone here on Earth, to
the motion of the moon, to the effects of electric currents on magnets, and to the
induction of a current in a solenoid are explained by postulating the existence of immaterial physical systems called fields. There are gravitational, electrical, and magnetic fields that fill space. They are said to be produced by electric charge, magnetic
charge, and by mass (which we may call gravitational charge). Since all objects
have mass, they are sources of gravitational field: An apple, the Earth and the Sun,
and fields themselves are creators of gravity. Positive or negative electric charge is
the source of electric fields, and magnetic charges which never occur alone but only
as dipoles—a combination of a north pole and a south pole—create magnetic fields.
Whereas all bodies have a gravitational field, a body needs a net electric charge to
have an electric field. Positive and negative charge cancel each other in their effects.
The phenomena showing coupling of electricity and magnetism demonstrate that
36
PHYSICS AS A SYSTEMS SCIENCE
2.2 QUANTITIES AND MATHEMATICAL OPERATIONS
electric and magnetic fields occur together—they are like the two sides of a coin. It
is customary to speak of electromagnetic fields. Gravitational and electromagnetic
fields are (immaterial) objects with their own properties and with effects upon each
other and on matter. Probably the best know of these are the mechanical effects of
gravity and the wavelike transports called electromagnetic waves (light, microwaves, radio, x-rays). Just like material objects, fields possess and transport some
of the basic fluid like quantities we use to describe phenomena—energy, quantity
of motion, and heat. For example, light transports heat, quantity of motion (momentum), and energy. The study of properties and effects of fields is a major subject of
physics. We will encounter fields again to a limited extent when we study motion
in Chapter 6.
2.2 QUANTITIES AND MATHEMATICAL OPERATIONS
Viewed from the perspective of how simple electric circuits behave, they appear
rather similar to some hydraulic systems. Also, our basic language used to describe
phenomena shows similarities: We speak of objects having electricity (charge),
charge flowing into or out of objects, and electricity being more or less intense. To
strengthen the sense of similarity and to demonstrate the use of analogical reasoning
that ties electricity to fluids and vice-versa, the following description of a theory of
electric phenomena is structured in parallel to that of fluids in Chapter 1. To the extent that this is possible, it is almost a copy of the pertinent parts of Chapter 1. Naturally, there are differences between fluids and electricity, and these will be
mentioned where appropriate.
Just like in hydraulics, we need three primitive system and process quantities to describe and explain electric phenomena. One is for amounts of electricity stored in
systems, the second for flows, and the last is for the electric potential at a point of
an electric system. On the basis of these, related ones are defined by mathematical
procedures. Properties of systems and elements—such as resistance and capacitance—are introduced together with special laws found to hold for the particular
system (see Section 2.6).
The role of energy in electricity (and for gravity and fluids) will be described only
briefly since there is an entire chapter devoted to energy (see Chapter E).
2.2.1 Primitives
Primitives are terms or quantities that cannot be defined on the basis of other quantities. They are fundamental and are taken from everyday notions and mental images of what we see happening around us.
Quantity of electricity: Charge. Electric phenomena suggest that we should introduce a quantity that measures an amount of electricity which is called electric
charge. The unit of electric charge is called the Coulomb (C). Remember that electric charge can be either positive or negative, and positive charge cancels the effects
of negative charge (the sum of equal amounts of positive and negative charge is zero). The charge of a system can change from positive to negative (and back) in the
course of time (Fig. 2.26). Electric charge can neither be produced nor destroyed.
Electric charge is the source of electric fields which are immaterial objects with par-
PHYSICS AS A SYSTEMS SCIENCE
37
Q
t
Figure 2.26: Charge (symbol Q) as a function of
time. Note how it changes: changes may be positive or negative, slow or fast. Charge can be positive or negative (as opposed to quantities of a fluid).
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
IQ
IQ
t
t
Figure 2.27: Currents of electric charge as a functions of time. Currents can change slowly or quickly,
they can be positive or negative (this has nothing to
do with positive or negative charge).
Figure 2.28: A traditional ammeter (left) and a
mordern multimeter (right).
ticular properties that affect other fields and matter. Most simply, fields explain interactions that appear to take place at a distance (without the direct interaction of
material objects).
Interestingly, electric charge is quantized: There is a smallest quantity of charge in
nature called the elementary charge. This smallest quantity is the amount possessed
by protons and electrons, for example. Protons carry one positive elementary
amount of charge, whereas the charge of an electron is one negative elementary
amount. The elementary charge is 1.60·10–19 Coulomb. This means that one mole
(one unit of a chemical species, see Chapter 4) of electrons or protons has a charge
of 96,485 C (this number is called Faraday’s constant).
Electric charge can be measured by its effects. One of the simplest effect is mechanical attraction or repulsion. However, rather than using such phenomena, it is more
common to quantify electric charge in terms of charge transported by electric currents. Electric currents are measured more easily in common situations.
Current of electric charge. The current of charge (electric current, or simply current) describes the flow of electric through materials. The current allows us to calculate how much charge is transported in a given period of time. The unit of electric
current is C/s which has its own name—Ampere (A). We use the symbol IQ to denote a flow of charge. In processes, the current is a function of time (Fig. 2.27).
Since charge can be positive or negative, there is a point worth mentioning. If nothing more is said, a current of charge refers to the flow of positive charge. This current can be both positive or negative—a negative current of positive charge simply
means a transport in the direction opposite to the one that is called the positive direction. If we want to refer to the transport of negative charge, we can do so by simply reversing the sign of the current. So a transport of negatively charged electrons
in the positive direction is counted as a negative current, and a flow of electrons in
the negative direction corresponds to a positive current of charge.
To give an example, in a simple circuit with a battery, a lamp, and metal wires, positive charge flows from the positive terminal of the battery through the wire to the
lamp, through the lamp, back to the negative terminal of the battery, and through
the battery to the positive terminal (and so on…). In the metal wires, electrons are
transported in the opposite direction (from the lamp to the positive terminal). In the
battery, however, ions are transported which—depending upon the chemicals and
the reactions—may be positively or negatively charged. If there are positive ions
flowing, they flow from the negative terminal to the positive terminal through the
battery.
The strength of electric currents can be measured relatively easily. Traditionally,
the magnetic mechanical effect of electric currents has been used to build ammeters
(Fig. 2.28). Note that the electric resistance of an ammeter should be as small as
possible, otherwise the measuring device will change the properties of the circuit it
is placed into too strongly (the resistance of the ammeter must be added to the resistances of the other devices in the branch of the circuit). Modern ammeters typically are multimeters that also function as voltmeters (Fig. 2.28, right).
Electric potential. The electric potential measures the intensity of the electric state
of a system at a point (these systems can be material bodies such as wire, but also
electric fields). Its unit is V (Volt), the symbol used is ϕel. It is analogous to pressure
in fluids. Electric potential does not have an absolute zero point (in contrast to pres-
38
PHYSICS AS A SYSTEMS SCIENCE
2.2 QUANTITIES AND MATHEMATICAL OPERATIONS
sure). This means that only differences of electric potentials (and differences relative to a freely chosen zero point) are important. Electric potential differences are
called voltage (see below). By itself, positive electric charge will flow from points
of high to low electric potential, whereas electrons flow from low to high electric
potential.
Q
2.2.2 Derived quantities
Change of charge. The change of charge of a system over a period of time is de-
fined as the difference between the later and the earlier values:
t
∆Q1→ 2 = Q2 − Q1
(2.1)
Rate of change of charge. The rate of change of charge describes how fast the
charge of a system changes (Fig. 2.29). Symbols are dQ/dt or Q with a dot above it
(pronounced Q-dot). The unit of rate of change of charge is C/s. The rate of change
is determined graphically from the slope of tangents at points of the curve in the
charge-time diagram.
Transported charge. If we know a current of charge as a function of time (as in
Fig. 2.27), we can determine the transported or exchanged amount of charge with
this flow (Fig. 2.30, symbol Qe). Graphically, the transported charge corresponds to
areas between the curve and the time axis for an interval of time from, say, t1 to t2.
The mathematical operation is called integration of the current over time:
Q̇
t
Figure 2.29: Determining the rate of change of
charge by calculating the slope of tangents to the
Q(t) curve. The symbol Q with the dot above it (pronounced “Q-dot”) denotes the rate of change of
charge Q.
t2
Qe = ∫ I Q ( t ) dt
(2.2)
t1
In system dynamics tools, this computation is performed automatically if one represents the current by a flow symbol connected to a storage element (Fig. 2.30, bottom). The storage element obtains the integrated quantity. The simplest numerical
method implemented in typical system dynamics programs (called Euler’s method)
is shown in Equ. 2.3:
Qe ( t ) = Qe ( t − ∆t ) + I Q ( t ) ∆t
IQ
IQ
Qe
Qe
∆t
(2.3)
t2
t1
t
Figure 2.30: The charge transported with aan electric current is determined by calculating the area
between the IQ(t) curve and the t-axis. The same
operation in a system dynamics model (bottom).
t
Transported
charge
Current of charge
PHYSICS AS A SYSTEMS SCIENCE
39
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
From rate of change of volume to volume. If the rate of change of charge stored in
Q̇
∆Q
∆t
t
a system is known, we can calculate the change of charge occurring during a period
of time (Fig. 2.31). As in the case of the computation of transported or exchanged
quantities (as a consequence of a flow), the mathematical procedure is that of integration, in this case of the rate of change of charge over time:
Q̇
t2
∆Q = ∫ Q ( t ) dt
∆Q
∆t
(2.4)
t1
t
Figure 2.31: Changes of charge of a system calculated from the rate of change.
If we also know the initial value of charge, we can calculate the charge as a function
of time (Fig. 2.32). This can be done graphically or numerically (in a spread sheet,
or with the help of a system dynamics tool).
Q̇
Q
∆Q
∆t
t
Charge
Qo
∆t
∆Q
Rate of change of charge
t
Figure 2.32: The charge of a system is found from the rate of change of charge and the initial value of charge by (numerical) integration. Right: Integrator in a system dynamics tool.
Voltage: the difference of electric potentials. The potential difference is the difference of potentials at two different points A and B of a system, independent of the
physical reasons for the difference. Usually, it is defined as later minus earlier in a
chosen direction:
∆ϕ AB = ϕ B − ϕ A
(2.5)
The negative potential difference is called the voltage between points A and B:
U AB = −∆ϕ AB
(2.6)
This means that the voltage is positive across a resistor in the direction of the flow
of (positive) charge, whereas it is negative across a battery in the direction of flow.
QUESTIONS
19. The lectric current through a resistor increases linearly from 0.10 A to 0.60 A in 20 s (as in the
diagram on the left of Fig. 2.27). How much charge has been transported in these 20 s? How
much charge is transported by a sinusoidal current in one full period?
20. The rate of change of charge of a capcitor is – 0.20 C/s. What is the charge of the capacitor at t
= 10 s if it was equal to 3.0 C at t = 0 s?
21. There are two points in the diagram on the left of Fig. 2.32 where the rate of change of charge
of a system equals zero. What happens to the charge of the system at those points?
22. There is a voltage of 3.0 V along a uniform wire of 0.50 m length. What is the gradient of the
electric potential along the wire in the direction of flow of charge? What is the sign of U?
40
PHYSICS AS A SYSTEMS SCIENCE
2.3 SYSTEMS ANALYSIS I: BALANCE OF CHARGE
2.3 SYSTEMS ANALYSIS I: BALANCE OF CHARGE
As in hydraulic systems (Chapter 1), there are two main steps to be taken to prepare
the modeling of an electric system. The first, the identification of electric currents
and of laws of balance will be discussed here. The second step consists of the identification of voltages (Section 2.4).
Fundamentally, step one is equivalent to what it is in a hydraulic system. We need
to find the objects that store electric charge and then identify all the transports of
charge with respect to them. Finally, this leads to expressions for the balance of
charge for every such object.
In a circuit having two capacitors as in Fig. 2.33, there are two storage elements.
There is a single current with respect to the first. (Remember, that an electric capacitor actually consists of two parts that store equal amounts of charge of opposite
sign, and there is the same flow into the system as out of it. Therefore, only one of
the parts of a capacitor has to be considered.) On the other hand, there are two currents leading into or out of the second capacitor. One of the currents is equal to the
one leading into or out of the first capacitor. (The wire or resistive element connecting the capacitors leads to an expression of interaction: The current into capacitor 2
the opposite sign of the one leaving capacitor 1.)
In system dynamics diagrams, laws of balance in their instantaneous forms can be
represented graphically by combinations of storage and flow symbols (Fig. 2.34).
Q1
IQ1
C1
IQ2
R1
C2
R2
Figure 2.33: Sketch of an electric circuit having two
capacitors and two resistive elements.
Q2
I Q1
Figure 2.34: Representation of laws of balance of
charge for the circuit of Fig. 2.33 in a system dynamics diagram.
I Q2
For each of the subsystems and for each of the stored quantities, a law of balance is
formulated (normally in dynamical or instantaneous form). A law of balance relates
all processes to how fast the system content changes:
Q1 = − I Q1
Q 2 = I Q1 − I Q 2
(2.7)
This is the form of the laws of balance applicable to the system of Fig. 2.33. It includes definitions of signs of currents and the interaction rule (the magnitude of the
current leaving the first capacitor equals the magnitude of the current entering the
second capacitor).
Many applications in electricity use circuits that do not have real storage elements
leading to dynamical processes (in Fig. 2.33, the capacitors are responsible for the
dynamics of the circuit). Fig. 2.35 shows the diagram of a circuit that leads to
steady-state processes. Charge is not stored anywhere. However, there still is an element for which a law of balance of charge has to be expressed. This is the junction
in Fig. 2.35 for which we write the junction-rule (Kirchhoff’s first law):
I Q1 = I Q 2 + I Q 3
PHYSICS AS A SYSTEMS SCIENCE
(2.8)
41
IQ1
R1
+
IQ2
Junction
IQ3
R3
R2
Figure 2.35: Electric circuits typically have juctions
(where wires split or join).
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
There is a second junction in the circuit of Fig. 2.35. However, it is equivalent to
the first leading to an equation equivalent to Equ. 2.8. Therefore, only a single expression of the balance of charge applies to this circuit.
QUESTIONS
23. The second capacitor in Fig. 2.33 receives a quantity of charge of 2.0 C during a certain period
from the first capacitor. During the same period, it loses 3.0 C through the second resistor. What
are the changes of charge of the capacitors (separately)?
24. Use an explicit expression for the interaction rule for the capacitors in Fig. 2.33. What is the
form of the laws of balance of charge for the capacitors in this case?
25. One often introduces a third electric current in the diagram of Fig. 2.33 leading from the junction to the second capacitor. Call this current IQ3. Rewrite the laws of balance of charge for both
capacitors. Is the system of equations still equivalent to what was formulated above?
26. Sometimes the junction rule (as in the example of Equ. 2.8) is formulated as follows: The sum
of all currents meeting at a junction is equal to zero. How does this agree with Equ. 2.8.
27. Combine the capacitors in Fig. 2.33 into a single one (for this to make practical sense, one
would make the resistance of R1 small compared to that of R2). Rewrite the law(s) of balance
for this case.
2.4 SYSTEMS ANALYSIS II: POTENTIALS AND VOLTAGES
C
B
+
D
F
E
G
UGH
A
Electric potentials takes the role of electric levels. Potential differences are level
differences which we consider to be the causes for electric processes. Alternatively,
we may look at processes leading to potential differences. Remember that a (negative) potential difference is called a voltage.
Potential differences (voltages) in closed electric circuits. The potential changes
from point to point in a closed electric circuit. To make use of this observation,
choose a few important points in the system such as the circuit of Fig. 2.36. Label
potential differences (voltages) from point to point by arrows and by symbols UAB,
etc. (Fig. 2.36).
We can draw a diagram looking like a “landscape” (Fig. 2.36). When we are back
at the origin, the potential is the same. Therefore, the sum of all voltages in a closed
circuit must be equal to zero (loop rule, Kirchhoff’s Second Law):
Potential
H
A B C D E F G H A
Position
Figure 2.36: Electric potential as a function of position in a circuit containing two capacitors, a battery, and a resistor. The values will change in the
course of time, but the form of the diagram will basically remain.
U AB + U BC + U CD + … = 0
(2.9)
Potential differences and processes. Potential differences (voltages) are associated
with many different types of processes and systems (some of which will be discussed in Section 2.6):
• charge stored in capacitors
• batteries and generators
• electric engines
• flow resistance
• electronic switches
• changing flows (changes in time will be discussed in a later chapter).
42
PHYSICS AS A SYSTEMS SCIENCE
2.5 ENERGY IN ELECTRICAL PROCESSES AND SYSTEMS
Processes in an element. In modeling electric circuits, real devices such as resistors,
batteries, capacitors, or cables are typically considered “pure” or ideal elements that
only exhibit a single property, namely the one they are named for. Thus a resistor
is modeled to only have resistive properties, a battery is assumed to be ideal (no production or heat), etc.
In reality, however, no element can be “pure” or ideal which means that more than
a single process proceeds in an element. A capacitor also has resistive properties, a
battery produces heat in addition to setting up a driving voltage, and a wire can also
have so-called inductive properties (this will be studied in Chapter 5).
In practice, it is only possible to measure a voltage across a real device such as a
battery (UAB in Fig. 2.37). To explain how the element operates, however, we have
to consider all processes taking place inside, and each of these is modeled to be associated with its own potential difference (voltage).Therefore, we introduce two or
more voltages making up the one that is measured across the device:
U AB = U process 1 + U process 2 + …
B
Ri
UAB
V
IQ
UR
+
Uo
A
Figure 2.37: The voltage across a battery is modeled as consisting of two terms, each associated
with a different process.
(2.10)
It is important to distinguish between the voltage across a device exhibiting more
than one process, and the voltages associated with each process in a model of the
device. The latter cannot be separated and measured independently. There is no
“pure engine” inside a battery setting up the voltage Uo coupled to a “pure” resistor
across which we have a voltage UR. The elements identified in the model do not exist independently.
QUESTIONS
28. Trace the electric potential in a circuit consisting of a real battery and a lamp. Sketch a level
diagram such as that in Fig. 2.36 at the bottom. Identify voltages. What is the relation between
all the voltages?
29. An electric motor contains coils of wire, and the wire has resistive properties. Is the voltage
measured across the motor during operation equal to the voltage that is said to cause the production of heat?
2.5 ENERGY IN ELECTRICAL PROCESSES AND SYSTEMS
Apart from being conserved, energy has three main properties: It can be stored, it
can flow or be transported, and it can be released or used in processes (see Chapter
E). This is true for energy in electrical systems as well.
Energy in electrical processes. The waterfall image of processes applies (Fig. 2.38):
The rate at which energy is released or used in an electric process is proportional to
the product of voltage (potential difference) and electric current:
Pel = −∆ϕ el I Q
(2.11)
Pel is called the (electric) power of a process (the rate at which energy is released in
a voluntary process or used in an involuntary process). ∆ϕel is the electric potential
difference through which we have an electric current IQ. Since the negative potenPHYSICS AS A SYSTEMS SCIENCE
43
IQ
ϕ1
IQ
Pel
Pel
ϕ2
ϕ1
ϕ2
Figure 2.38: Process diagrams of voluntary (driving,
left) and involuntary electric processes (right).
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
tial difference is the voltage, a voluntary electric process is associated with a positive voltage and positive electric power (conversely, an involuntary electric process
such as the one driven by chemical reactions in a battery has negative power).
Energy transport with conductive electric transports. If electric charge flows (conductively) into or out of a system, and the flow is associated with a potential ϕel
(Fig. 2.39), there is an associated energy current IW,el of
System
IW
IQ
IW , el = ϕ el I Q
ϕ
Figure 2.39: Energy currents are associated with the
tranposrt of charge into or out of a system.
(2.12)
Energy storage. Energy can be stored in electrical systems together with electric
charge (in fact, it is said to be stored in the electric field associated with the electrical state of the system). The simplest example is a capacitor where the quantity of
energy stored is related to the charge and the voltage of the capacitor. The precise
relation depends upon the particular properties of the storage elements, so there is
no single relation that can be used for energy storage (see Chapter E for more information).
2.6 CONSTITUTIVE LAWS: CAPACITORS, BATTERIES, AND FLOW
Constitutive laws, or special laws, are relations that depend upon the types of elements used in a system, and upon circumstances. In contrast to the laws of balance,
the loop rule, or the expression for power, they are not general. They describe the
peculiarities of processes and objects.
There are constitutive laws covering all different types of processes. In particular,
there is at least one special law for each particular potential difference occurring in
a system.
2.6.1 Storage of charge in capacitors
UC
Q
UC
Q
UC = 0
C
Q
UC
Figure 2.40: Voltage as a function of stored charge
(top). The slope of the characteristic curve is called
the elastance of the capacitor. Fluid image of capacitors (bottom). A capacitor is like a tank storing
charge (Q) which can be positive or negative. The
level in the “tank” depends upon the quantity
stored through the cross section of the tank. Here,
the cross section symbolizes the capacitance.
Storage elements are responsible for the dynamics found in systems. They work by
providing a relation between amounts of stored charge and the voltage set up across
them. Storage elements are capacitors.
Capacitive characteristic. If charge is stored in a capacitor, the voltage increases
with increasing amount of stored charge. In other words, there is a relation between
the charge stored and the associated voltage (which we call a capacitive voltage
UC). The relation is called a capacitive characteristic (Fig. 2.40, top). Another way
of representing the relation is by drawing a fluid image (Fig. 2.40, bottom), an
imaginary tank with charge inside where the level represents the voltage UC. In general, the characteristic is nonlinear. A linear characteristic is related to a constant
capacitance. The cross section of the imaginary tank represents the capacitance of
the capacitor.
Elastance and capacitance. The characteristic relation can be expressed mathematically if we introduce the elastance αQ, i.e., the factor which tells us how easy
it is to increase the voltage with a given amount of charge:
U C = α QQ
U C = α QQ if α Q = const .
44
PHYSICS AS A SYSTEMS SCIENCE
(2.13)
2.6 CONSTITUTIVE LAWS: CAPACITORS, BATTERIES, AND FLOW
αQ is equal to the slope of a tangent to the characteristic curve (Fig. 2.40). This
means that the elastance measures the “stiffness” of the storage system. The unit of
elastance is V/C.
Alternatively, we can introduce the electric capacitance CQ or simply C (units C/V
= F (Farad)) which is defined as the inverse of the elastance (C = 1/αQ):
Q = CQU C
Q = CQU C if CQ = const .
(2.14)
Energy stored in capacitors. The fluid image of a capacitor in Fig. 2.40 lets us cal-
culate the energy stored with charge in a capacitor. The charge stored is represented
as the content of the imaginary tank. The equation for the energy associated with
this storage is the same as the one for the energy necessary to stack a real fluid in a
real tank here on Earth. For a capacitor having constant capacitance, stacking
charge in the tank is like placing it all at half the total height. Therefore:
WC =
1
CU C2
2
(2.15)
QUESTIONS
30. What is the electric current through an immersion heater hooked up to 220 V having an electric
power of 300 W? What is the thermal power of this device?
31. Consider a capacitor having constant capacitance. What is the form of the capacitive characteristic?
32. Imagine a capacitor being discharged in a simple circuit. What kind of data should be taken to
derive the capacitive relation of the capacitor? How do you determine it?
33. Explain the meaning of electric capacitance.
34. Consider a capacitor made of two parallel metal plates. What happens to the capacitance if the
surface area of the plates is doubled?
35. A capacitor having a capacitance of 100 µF (micro-Farad) is charged to a voltage of 10 V. How
much charge and energy is stored (on one of the plates in a parallel plate capacitor)?
2.6.2 Batteries
Batteries or generators (as large as those in a power plant or as small as photovoltaic
or thermoelectric generators, see Chapter 3) are used to set up voltages to drive electrical processes. Here we only describe batteries.
Process diagram. Batteries make electric charge flow, and they increase electric potential of charge. This simple fact is best represented in a process diagram of the
type shown in Fig. 2.41. Process diagrams represent a system (or an element of a
system) and show what happens with the basic quantities (here: current of charge,
voltage, and energy related quantities) used to describe a electric process. In a real
battery, a part of the energy released by the chemical reaction is used to produce
heat (as long as there is an electric current in the circuit with the battery). This reduces the voltage set up by the chemical reactions.
Characteristics of batteries. Characteristic diagrams of batteries are simple linear
curves: The voltage across the terminals decreases with increasing electric current
PHYSICS AS A SYSTEMS SCIENCE
45
Battery
Pth
E
Heat
IQ
Pch
Pel
ϕ1
ϕ2
Figure 2.41: Power, energy currents, and energy
storage are associated with the operation of a battery. The voltage measured across the terminals
(here: ϕ2 – ϕ1) is smaller thatn the maximum possible value determined by the chemical reactions.
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
(see Fig. 2.11). The formal description of the characteristic is
U B = U 0 − Ri I Q
Current / A
4
3
[
[
2
[
P
[
1
[
P
[
0
0
1
2
3
Voltage / V
[[
[
[
[
4
5
Figure 2.42: Determination of the maximum power
of a battery in operation.
(2.16)
UB is the voltage measured across the terminals, U0 is the maximum possible voltage (the open circuit voltage), IQ is the current through the battery, and Ri is the internal resistance of the battery (Fig. 2.12). A derivation of Equ. 2.16 uses ohm’s
relation for resistive charge transports (see Section 2.6.3).
Maximum power of a battery. Fig. 2.42 shows gain the characteristic diagram of a
standard battery. The voltage across the terminals decreases if the current increases
(this is the case if we make the resistance of a load resistor smaller as was done for
getting the data of Fig. 2.42).
The useful power of the battery, i.e., the rate at which energy is supplied to the electric process driven by the battery, is determined by the product of the voltage across
the terminals and the current through the battery. In the characteristic diagram, this
product equals the area of a rectangle below the characteristic curve (see the two
rectangles in Fig. 2.42). Graphical inspection shows that this area is small if either
the voltage is high or the current is high. There is a maximum of the power somewhere in between. The maximum can be determined either numerically or analytically from Equ. 2.11 and Equ. 2.16. The maximum is achieved when the resistance
of the load resistor is made equal to the internal resistance of the battery.
2.6.3 Resistive charge transport
Resistor
Pth
IQ
ϕ1
Heat
Pel
ϕ2
Figure 2.43: Process diagram of resistive fluid flow.
The flow element may be called a resistor. Here, the
driving process is the flow. The driven process consists of the production of heat. Right: Waterfall representation.
When electric charge flows through what we normally call resistors or conductors,
heat is produced. This is the main characteristic of the type of transport we are considering here. Obviously, there are transports that are not resistive such as the main
phenomenon in a battery or the part of the flow of electricity through a motor that
drives the engine.
Process diagram. Charge flows through a resistor (or a conductor) from higher to
lower electric potentials which means that energy is released as a result of this transport. The energy released is used to produce heat (Fig. 2.43).
Resistive characteristic. The relation between the resistive voltage UR and the associated current of charge is called the resistive characteristic (Fig. 2.8 and Fig. 2.9).
It allows us to calculate flows of charge if we know the associated voltage, or viceversa. There are basically two types of transport called ohmic and non-ohmic leading to two different characteristic curves.
Ohmic transport of charge. The transport of charge in metallic conductors satisfies
a simple relation. For small enough voltages or electric currents, the current is strictly proportional to the potential difference across the conductor. Therefore, the characteristic relation is linear. In this case, we can write the flow law with the help of
a conductance G (units A/V = 1/Ohm) or its inverse, the resistance R (V/A = Ohm
= Ω):
I Q = GU R or I Q =
1
UR
R
(2.17)
There is an expression for the conductance or resistance for ohmic transport in conductors having constant cross section:
46
PHYSICS AS A SYSTEMS SCIENCE
2.6 CONSTITUTIVE LAWS: CAPACITORS, BATTERIES, AND FLOW
R = ρel
l
A
(2.18)
l and A are the length and cross section of the conductor, respectively, and ρel is the
resistivity of the material. The resistivity basically measures how hard it is for
charge to flow through the conductor. The inverse of the resistivity is called the
electrical conductivity: σ = 1/ρel (unit: S/m, S: siemens). If we introduce Equ. 2.18
into Equ. 2.17, we obtain IQ = AσUR/l. For the homogeneous conductor, IQ/A is the
current density of charge jQ; UR/l is the gradient of the electric potential. This suggests that
jQ = −σ
dϕ el
dx
(2.19)
djel/dx is the gradient of jel which measures how fast the potential is changing in the
direction of flow of charge. Equ. 2.19 is the most general expression for ohmic type
of conduction of charge; it is called Ohm’s law. Since σ may depend upon the temperature of a conductor, Ohm’s law may lead to nonlinear characteristic curves as
for the filament of a light bulb (Fig. 2.8, right). The important aspect of Ohm’s law
is that the transport of charge is proportional to the gradient of the potential.
The temperature dependence of the conductivity or the resistivity is commonly approximated by a linear (or if necessary a quadratic) function of temperature. The coefficient α multiplying the linear term is called the (linear) temperature coefficient
of resistivity:
ρel = ρel ,20 (1 + α (T − T20 ))
(2.20)
Series and parallel connections of ohmic resistors. If ohmic resistors having con-
stant resistances are combined in circuits, series and parallel combinations can be
replaced by equivalent resistors. For resistors in series (Fig. 2.44), the equivalent resistance is the sum of the individual resistances:
Requiv = R1 + R2 + …
Series connection:
IQ
R1
R2
(2.21)
Parallel connection:
If several resistors are connected in parallel (Fig. 2.44, bottom), the inverse of the
equivalent resistance equals the sum of all inverse values of the individual resistances:
IQ
R1
R2
1
Requiv
1
1
=
+
+…
R1 R2
(2.22)
Power of an ohmic resistor. The electric power of a resistor is calculated from the
standard expression Equ. 2.11 and from Equ. 2.17 and can be expressed either with
voltage or with the current:
Pel , R =
1 2
U R = RI Q2
R
(2.23)
Transport of charge in diodes. An example of a diode characteristic is seen above in
Fig. 2.9. The current can be approximated by an exponential function of the voltage
PHYSICS AS A SYSTEMS SCIENCE
47
Figure 2.44: Resistors in series (top) and in parallel
(bottom).
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
UD across the diode:
⎛
⎛ U e⎞ ⎞
I Q = I 0 ⎜ exp ⎜ D ⎟ − 1⎟
⎝ nkT ⎠ ⎠
⎝
(2.24)
I0 is called the saturation current, n is an emission coefficient with a value of somewhere between 1 and 2 (depending on the material and the fabrication of the diode).
e is the elementary charge (1.60·10–19 C) and k = 1.38·10–23 J/K is Boltzmann’s constant. T measures the absolute temperature of the diode in Kelvin (Chapter 3).
Electrolytes. Electrolytes are electrically conducting solutions of chemicals. The
transport of charge is coupled with the transport of ionized chemicals. For example,
if normal table salt (NaCl) is dissolved in water, sodium and chlorine ions are
formed (Na+ and Cl–). If two electrodes (metal rods) connected to a power supply
are introduced in the salt water, an electric current is measure. In the salt water, the
transport of charge is due to the transport of the ions.
Electrolytes are important biologically and technically. Many cell functions—not
the least those of nerve cells—rest on the transport of ions across cell membranes.
Technical devices that make use of electrolytes are, among many others, batteries,
fuel cells, some types of capacitors, and aluminum smelters.
QUESTIONS
36. Why is the current through a battery limited, even for load resistors with very small resistances?
37. Why is there a maximum of the power of a battery in operation? What is its power for either
maximum voltage or maximum current?
38. Estimate the maximum power of the photovoltaic array in Fig. 2.24 for an insolation of 400 W.
39. Imagine two identical resistors. What is their equivalent resistance if they are connected in parallel?
40. A 100 W incandescent light bulb is connected to 220 V. What is the resistance of the filament?
41. What is the ratio of the resistance of copper wires having the same length where one has double
the diameter of the other?
42. Derive the expressions for equivalent resistances of resistors in series or in parallel.
2.7 DYNAMICAL MODELS AND SYSTEM BEHAVIOR
Causal physical models are answers to the question “why:” Why is a system in a
certain state? Why do processes run a certain way? A complete model of systems
and processes is simply a combination of all relations—laws of balance and constitutive laws we have collected so far—necessary for a particular example. The purpose of a model is to determine quantities describing a situation at a moment, or to
predict the outcome of processes.
2.7.1 Dynamical models
Dynamical models combine laws of balance with the appropriate constitutive laws.
They are created by a combination of steps described above in Section 2.3 (Systems
analysis I: Laws of balance) and Section 2.4 (Systems analysis II: Potential differ-
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PHYSICS AS A SYSTEMS SCIENCE
2.7 DYNAMICAL MODELS AND SYSTEM BEHAVIOR
ences), with the particular laws for special processes found in Section 2.6 (Constitutive laws). Expressions for energy relations (Section 2.5) can be added to the
dynamical models.
If the model describes a dynamical situation, it may be expressed with the help of a
system dynamics tool. A system dynamics diagram represents the necessary laws
of balance and constitutive laws. The example of Fig. 2.45 is for an electric circuit
that is equivalent to a simple hydraulic windkessel model.
Tank
Q
Pump
IQ 1
Valve
IQ 2
R1
R1
R2
C
~
R2
U R1
U power
supply
U R2
UC
UD
Figure 2.45: A system dynamics model diagram for
a system equivalent to a hydraulic windkessel.
C
Mathematically speaking, the completed model is a set of equations that have to be
solved during simulation. The equations comprise a single differential equation for
the law of balance of charge of the capacitor. Then there are equations for the relations between voltages in the two branches of the circuit. Finally, there are expressions for the capacitance of the voltage of the capacitor, and for the electric currents.
2.7.2 Analytical solutions
Systems made up of capacitors and resistors show relatively simple behavior. Complex behavior is commonly the result of the interaction of several simple elements.
For the simplest systems—those having constant values of capacitance and resistance—analytic solutions of the model equations can be obtained. In the case of discharging a constant capacitance capacitor through an ohmic resistor we get
U C ( t ) = U Co e
−
t
RC
(2.25)
If an empty capacitor is charged, the solution of the model is
t
−
⎞
⎛
UC ( t ) = U max ⎜ 1 − e RC ⎟
⎠
⎝
(2.26)
These results also hold for the charge stored in the capacitor. We simply multiply
the equations by the capacitance to obtain the new results.
2.7.3 Time constants
The behavior (capacitive voltage as a function of time) for the simple cases of
charging and discharging of a capacitor is shown in the accompanying graphs. The
PHYSICS AS A SYSTEMS SCIENCE
49
CHAPTER 2. ELECTRICAL SYSTEMS AND PROCESSES
solutions of the model are exponential functions. A measure of how fast (or slow)
the process is, is the time it would take for the capacitor to charge or discharge were
the voltage to continue to change at the initial rate. This time is called the capacitive
time constant τC of the system. In one time constant, the capacitive voltage in the
system shown on the left in Fig. 2.46 drops to 1/e = 0.37 times the initial level. The
analytic solutions in Equ. 2.25 and Equ. 2.26 demonstrate that
τ C = RC
(2.27)
UC
R1
+
C
R2
V
Figure 2.46: Discharging or charging of capacitors
having constant capacitance through resistors having constant leads to exponentially changing functions. The initial rate of change is used to define the
time constant of the system.
UC
τC
t
τC
t
UC
QUESTIONS
43. In what sense can we say that a model such as the one in Fig. 2.45 explains a system?
44. Explain the meaning of the structure of stocks (rectangles) and flows (thick arrows) in the system dynamics diagram of Fig. 2.45. Why is there only a single stock? Why are there two flows
connected to the stock for Q?
45. In Fig. 2.45, UD symbolizes the voltage across the diode which we set to a constant value independent of the current. How is UR1 calculated?
46. To what percentage of the final level does the level on the right in the diagram of Fig. 2.46 rise
in one time constant?
47. Estimate the time constant of the circuit in Fig. 2.2.
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PHYSICS AS A SYSTEMS SCIENCE