Electricity and gravity are the most fundamental interactions that we commonly
experience. In the same way that mass figures into gravitational forces, so too charge is
involved in electrical forces. However, as we'll soon see, gravitational and electrical
forces differ in one major respect: while a gravitational force can only be attractive, an
electrical force can be either attractive or repulsive.
Charge and Coulomb's Law
In 1785, the French scientist Charles Coulomb (1736–1806) found that the electric force
between two charges Q1 and Q2, separated from each other by a distance r, is given by
[Coulomb's Law] F=k·Q1·Q2/r2
where k=9x109 N·m2/C2 is the Coulomb constant, often expressed in terms of another
constant as 0=1/4πk=8.85x10–12 C2/(N·m2). The direction of the force between two
charges is always along the line connecting them. If several charges are present, then the
net force on any one charge is the vector sum of the forces due to all the other charges.
It's interesting that Coulomb's law very much resembles Newton's law of
gravitation. However, since charge can be positive or negative, the force in Coulomb's
law can be either attractive or repulsive. This fact is expressed formally in the law of
[Law of charges] Opposite charges attract, like ones repel
Example: Richard Feynman once said that if two people stood at arm’s length from each
other and each had 1% more electrons than protons, the force of repulsion between them
would be great enough to lift the weight equivalent of the Earth. Was he right?
Solution: Water has a mass of 18 g/mole, where 1 mole=6.02x1023. Assuming a 70-kg
individual is composed mostly of water, we estimate the number of water molecules in
his/her body is 70,000/(18)=3889 moles=2.3x1027 molecules. Since water has 10
protons per molecule, this normally corresponds to Np=2.3x1028 protons and an equal
number of electrons. A 1% charge difference amounts to 0.01·Np=2.3x1026 electrons.
Assuming ―arm’s length‖ means about 0.5 m, the repulsive force would be
9x109·(2.3x1026·1.6x10–19)2/(0.5)2=4.9x1025 N. The Earth has a mass of M=6x1024
kg and therefore weighs M·g=6x1025 N. It would seem, therefore, that Feynman was
certainly in the ballpark.
In the late 1700s, Benjamin Franklin arrived at the conclusion that the excess or
deficiency of a "charging" fluid causes an object to exhibit the electrical properties of one
of two possible states. He arbitrarily assigned rubbed glass to have an excess of this fluid
and therefore called its charge state positive. On the other hand, he believed that rubbed
rubber had a deficiency of this fluid and was therefore to be considered as negatively
charged. In his view, this fluid could be exchanged between objects by friction. From
today's point of view, Franklin's convention is somewhat of a misnomer because what he
considered to be a deficiency of a fluid is now known to be an excess of electrons.
Nevertheless, Franklin's convention persists to this day, as do many other historical relics.
Answer: It is actually the presence or absence of negatively charged electrons, and not of
a positively charged fluid, that defines the charge of an object.
The Amber Effect
Hermes and many other Greek divinities were accustomed to swift and unimpeded
passage through space. Wearing their characteristic talaria (winged sandals), these gods
soared even more impressively than Michael Jordan accoutered with his Air Nikes. The
Greek mortals, however, were much less adept at defying gravity, with or without
auxiliary footwear. Nevertheless, the Greeks did manage, though on a much smaller
scale, to produce movement and even levitation in small bits of matter. They did so by
first rubbing pieces of amber together and then using the "energized" amber to attract
other small objects. This charging phenomenon is sometimes known as the amber effect
or more formally, in more modern terms, as triboelectricity (in Greek, tribo- means rub
elektron means amber).
Is gravity really defied here? Well, in a sense. Although gravity is still present, a
much stronger force—the electric force—predominates. Rubbing objects together can
actually lead to the transfer of charge from one object to the other. In modern times, the
effect should be familiar to anyone whose hair comb is suddenly transformed into a
"paper magnet," capable of lifting small pieces of paper, after only a few strokes of the
hair. In this case, the comb becomes positively charged by giving up some of its
electrons to the hair. When the positively charged comb now comes close to the neutral
paper, it polarizes (i.e., induces a slight charge separation in) the paper (which remains
neutral) and attracts the negatively charged side.
The amber effect can be understood in terms of a property of atoms and molecules
called electron affinity, the ability of an atom or molecule to hold on tightly to its outer
(valence) electrons. Materials with low electron affinities tend to become positively
charged when rubbed against or even simply coming into contact with higher-affinity
materials. The table below summarizes the relative electron affinities of several typical
materials. We see, for example, that fur has a greater electron affinity than does glass;
consequently, a piece of fur will become negatively charged when rubbed against a glass
rod. Of course, the glass rod, which gives up the electrons, will become positively
charged in the process. A typical number of electrons transferred during such a rubbing
event is around 109, a number reminiscent of the national debt.
Table: Relative electron affinities
of several common materials
Human skin, aluminum
[Relative electron affinities of several common materials, with affinity increasing
downward in the table. Upon contact between any two materials in the table, the material
appearing above is expected to become positively charged (i.e., loses electrons), while the
one listed anywhere below it becomes negatively charged (i.e., gains electrons).]
Question: An illustrative example of triboelectricity is the lightning which is sometimes
observed during large desert sandstorms, even in the absence of thunderclouds. What
possible connection could sand have to lightning?
Answer: Evidently, some charge separation takes place as a result of friction of sand
blowing hard against sand. The eventual recombination of the charge produces either
sparking or, on a larger scale, lightning.
Example: Suppose a glass rod gives up 109 electrons when rubbed against a silk cloth.
What fraction of all its electrons does this glass rod give up? Assume the glass rod has a
mass of 120 g and is composed mostly of silicon dioxide (SiO2).
Answer: The atomic mass (number) of Si is 28 g/mole (14) and that of O2 is 32 g/mole
(16), thus 60 g/mole of SiO2. Thus, the glass rod contains 120/60=2 mole of SiO2
molecules, equivalent to about 30·2·6.02x1023=3.6x1025 protons or electrons. The
fraction represented by 109 electrons is 109/3.6x1025=3x10–17, a tiny fraction.
A Matter of Perspective*
Is an electric force really much stronger than the gravitational force? Well, that depends
on your perspective. Certainly, the electric force between an electron and a proton is
enormous compared with the gravitational force between the two particles. In fact, the
ratio of the two forces is around 1039 (see example below). But why do we not sense
such enormous forces? If gravity is great enough to keep us attached to the Earth, should
we not be crushed by the much larger electrical forces? Fortunately for us, nature has
created positive and negative charges in equal numbers. Thus, the electrical forces that
we usually feel arise between essentially neutral objects and are therefore quite negligible.
Alternatively, we can picture large attractive forces being completely canceled by large
repulsive ones. When the cancellation is not quite complete, then we get a small taste of
the potentially huge electric force.
We can generalize by saying that on the atomic scale Coulomb forces are much
more important than gravitational forces. As we ―zoom out‖ and enter the macroscopic
world, however, gravitational forces become more important because mass grows just as
the objects do, while the net charge stays very small. Thus, the Coulomb force between
the Sun and the Earth is quite negligible, while the graviational force between these two
celestial bodies keeps us in orbit and protects us from the unenviable plight of wandering
aimlessly in space.
Example: Estimate the ratio of the Coulomb force between an electron and a proton to
the gravitational force between the two entities.
Solution: The Coulomb force is given by k·e2/d2, while the gravitational force is given
by G·me·mp/d2. The ratio of the two forces is therefore
Waking up to a Cold Shower
Taking a shower always produces a refreshing feeling. But is there more to it than just
feeling clean? In fact, the splashing water in the shower can ionize the air in the
bathroom and produce electric fields up to 800 V/m. Nowadays an item called the ionizer
can be found in many electronic or department stores. This device generates negatively
charged particles (anions) in the air that you breathe. Specifically, it generates negatively
charged oxygen molecules through the following reaction: O2 + e– O2–. This
reaction also occurs naturally in the atmosphere. However, its effects are partly
neutralized by another reaction CO2 CO2+ + e– which produces positively charged
ions (cations). In recent years, it has been thought that the ion density in the atmosphere
may be an important factor in our general well-being.
In a suburban or rural area, especially in a forest, people often claim the sensation
of breathing easier. Certainly, the air smells fresher. An important reason behind these
perceptions may be that the O2– concentration in the air is considerably higher than that
in a congested urban setting, such as the inside of a high-rise office building in downtown
Los Angeles. In fact, there are typically a few thousand O2– anions per cubic centimeter
of air in a forest, while the number of O2– anions in an enclosed office space can dwindle
to only a few per cubic centimeter of air. Although the mechanism behind the influence
of O2– anions on our health is not yet understood, the possibility that they do seems to be
supported by quite a bit of circumstantial evidence. Breathing in O2–-rich air can boost
our immune systems, help prevent or cure a number of diseases such as hypertension,
insomnia, asthma, and even malignant tumors. No wonder these air cleaners/ionizers are
making their way to a lot of hospitals, offices, and homes.
Attracting an Image
A typical photocopier has a drum made of a conductor (aluminum), which is coated with
a thin layer of a semiconductor selenium. In operation, the selenium layer is first charged
by a spray of charged air molecules. In the dark, selenium is a poor conductor, and the
charges remain in place. Light is reflected from the white parts of the page to be copied,
passed through a lens, and focused on the drum. Where the light strikes the selenium, the
semiconductor becomes a conductor, letting the charges flow away from the surface to an
aluminum drum. Dark areas on the page, however, correspond to charged areas on the
selenium layer. The drum is then rotated through a container of toner, consisting of tiny
charged plastic beads coated with carbon grains, which are attracted only to the charged
areas of the selenium layer on the drum. The coated beads are then transferred to a sheet
of paper as it is pressed against the drum. The paper is heated and the beads melt,
attaching the carbon to the paper to form the image.
Salt and Paper—Demo
What do salt and paper have in common? Recall from chemistry that salt is composed of
sodium and chlorine ions held together in a crystal by electrostatic forces. Paper is
composed of long cellulose fibers also held together by electrostatic forces. When salt is
put into water, it quickly dissolves and breaks up into individual ions. By surrounding
and shielding the individual ions, water weakens the interionic attraction. In a similar
manner, water weakens the cohesive electrostatic forces between the cellulose fibers.
Wet paper is not only noticeably weaker than dry paper, it also cannot produce the crisp
Faraday’s Ice Pail—Demo*
When a net charge is placed on a conductor, the charge distributes itself in such a way
that the electric field inside the conductor is zero. Many experiements have been
performed to show this. In one such experiment, a positively charged metal ball at the
end of a silk thread is lowered into an uncharged, hollow conductor through a small
opening. Faraday used an ice bucket for a conductor. When the ball touches bottom, it
becomes part of the conductor. When the ball is removed, all of its charge remains on the
conductor. Charge on the conductor is measured by an electrometer, whose reading does
not change as the metal ball is being removed.
Can your skin really crawl? Of course not! But it can sure feel like it. All you have to do
is take a woolen sock and rub it for several minutes against a styrofoam plate. Then,
position the plate vertically next to a friend’s arm and slowly move it back and forth. (A
hair comb just after combing could serve the same purpose.) Since the plate has
accumulated a substantial charge from the rubbing, it induces an opposite charge on the
hairs. The movement of the hairs as they follow the plate produces a sensation of ants
crawling on your skin. Appropriately, this sensation is called formication (from the
Latin formica, meaning ant). They just don’t teach you stuff like this in Sunday school!
A Bad Hair Day—Demo*
The Van de Graaff generator is a device capable of transferring a lot of charge onto a
metal terminal (usually, a spherical shell). A person touching the terminal becomes
electrically charged. As the hairs on the person’s head repel each other, they tend to
follow the electric field lines.
Charging by Induction
For Whom the Bell Tolls
Stormy weather always tends to create a certain mood. Some people, for instance, claim
to hear musical tones periodically. Is there some physical explanation or is this yet
another excerpt from the files of delusionary phenomena? Interestingly, Benjamin
Franklin once wrote that he had "erected an iron rod to draw the lightning down into my
house, in order to make some experiment on it, with two bells to give notice when the rod
should be electrify'd...." One bell was actually connected to the iron rod so that it could
be charged, while the other was grounded (i.e., attached to the Earth). A small metal ball
was suspended on a silk thread so it could swing back and forth between the two bells,
ringing when struck. Every time lightning would strike, it would charge the bell
connected to it. The suspended metal ball, initially neutral, is then attracted to the
charged bell by induction. After touching the bell, the ball picks up some charge and then
swings to the other bell and gets discharged. As long as lightning keeps striking, the
chiming continues. So for whom does the bell toll? Perhaps for Benjamin Franklin.
Dressed to Kill*
If you are like most other people, you probably prefer natural fabrics (such as cotton,
wool or silk) for your clothing, especially those that are in direct contact with your skin.
(Indeed, few people would opt for nylon over 100% cotton for their underwear.) But is
there something more to it than just a matter of choosing a more comfortable fabric?
At least for some people, the answer is yes. There have been incidents in which a
person with no history of heart problems would suffer from heart discomfort, including
irregular beats, when he or she wears a certain type of synthetic fabric. These problems
would disappear soon after the person switches back to natural fabrics.
The cause for this symptom is very much similar to the case of precious-metal
syndrome. Only this time it is the synthetic fabrics that is causing the trouble. A synthetic
sweater, for instance, can easily pick up 15,100 volts. The resultant change in the
potential distribution in our body can sometimes cause disturbance to the flow of the
body current, which is vital to our well-being.
Have you ever noticed that television and computer-monitor screens love to collect dust?
As the screen becomes charged with electrons hitting it, dust particles are attracted to it
via the amber (―static‖) effect. Fortunately, the collection of dust can serve a purpose
beyond that of simply being a nuisance. The principle behind it is now routinely applied
to the control of pollution. Tiny particles of soot, ash, and dust are major components of
the airborne emissions from fossil-fuel-burning power plants and from many industrial
processing plants. The so-called electrostatic precipitators can remove nearly all of these
particles from the emissions. The polluted flue gas is passed through a series of charged
metal plates and negatively charged wires. The strong electric field around the wires
creates negative ions in the particles, which are attracted to and collected by the positively
charged plates. The plates are periodically shaken in order to cause the polluting particles
slide down into a collection hopper. The seemingly worthless ―fly ash‖ can be either
disposed of or sometimes used as filler in concrete.
Just a few years ago, chains or wires were commonly dragged along the road surface from
the bodies of trucks. Their purpose was to discharge any potential charge that might
potentially build up because of friction with the air. Nowadays, electrically conducting
tires are commonly used to prevent charge build-up, especially in trucks carrying
flammable cargoes. At some automobile toll-collecting stations, a thin metal wire sticks
up from the road and makes contact with cars before they reach the toll collector. In
operating rooms, where flammable anesthetics are used, the floors are made of
conducting material and the doctors and nurses wear conducting footwear. Without these
precautions—if charge were allowed to build up—the inevitable spark would result in
something other than an honorable discharge!
Electric Field and Potential
Electrical forces, like gravitational forces, act between objects which are not in contact
with each other. We can therefore imagine a force field to be present which influences
charges and masses, respectively. An electric field is defined as the force per unit charge
that a charge would feel in the presence of other charges. An electric field can be
represented by lines of force, which are always directed away from positive charges and
toward negative charges. The strength of the electric field is proportional the density of
+ Ğ +
Because field lines always originate from or converge toward some charge, one
might think that the number of lines per unit area passing through a closed surface
surrounding a charge should tell us about the magnitude and sign of the enclosed charge
Qin. In fact, this line of reasoning leads us directly to Gauss' law:
[Gauss' law] e=(E·∆A·cos)=Qin/0
where e is called the flux of the electric field through the surface, and is the angle
between the electric field lines and the normal (perpendicular) to a section A of the
surface. Keep in mind that Gauss' law applies only to a closed surface, in which case the
flux is equal to Qin/0. Otherwise the flux can take any value, which can be calculated
from the left part of the above equation. Furthermore, Gauss' law, while always true, is
useful only when there is enough symmetry in the problem to solve for E.
The electric potential V is related to electric potential energy in the same way as the
electric field is related to the electric force. Of course, potential energy must always be
expressed in terms of differences:
[electric potential energy] (EPE)=q·∆V
where ∆V is the potential difference, often called voltage. We can think of the voltage as
the work per unit charge required to move some charge across a distance ∆s in an electric
field E. If E is uniform, (constant) then ∆V=E·∆s. When single atoms or electrons are
involved, a convenient energy unit is the electron volt
[electron volt] 1 eV=1.6x10 –19 J
An Electrifying Kiss
The voltage supplied by domestic household outlets in the U.S. is only 110 Volts. You
might think that any voltage higher than that is not easy to get, at least not without some
electrical equipment. Not so. Get into your walk-in closet, find any dress (or suit) made
of some synthetic material, and give it a single rub. This will give you 100 volts. How
about several hundred volts? That's also not hard to get. All it takes is to walk a few
steps on any carpet made of artificial material. You can produce over 10,000 volts simply
in the process of quickly putting on or removing a sweater, something you probably do
every day. Such a seemingly dangerous voltage should not be too disturbing, since the
high voltage is not accompanied by a high current, which really would be fatal.
Fortunately, the relatively high resistance of our bodies prevents this from happening.
However, the omni-present static electricity can cause considerable damage to many
delicate pieces of electronic apparatus.
When you walk across a rug on a dry day, your body can pick up a lot of excess
charge and thus gain a large electric potential. The potential produces an electric field
around you. Since an electric field pulls opposite charges in opposite directions, your
body potential may ionize some of the air molecules around you. The ions can create an
electric conduction path, similar to that due to the salt in water. If the magnitude of this
potential is large enough, a spark can jump between your hand and a conductor, such as a
metal surface. If air can tolerate a maximum electric field of 3 kV/mm before breaking
down (ionizing), how close can you get to another person without creating a spark when
your body is charged to a potential of 9 kV? Well, ∆s=∆V/E=3 mm. It's no wonder that
sparks sometimes fly when you kiss someone!
Example: How many electrons pass between the terminals of a 12-V car battery when a
60-W headlight burns for an hour?
Solution: In one hour, the headlight uses an energy of E=60·3600=2.2x105J. Since this
energy is provided by the battery, E=q·∆V or q=1.8x104 C, which is equivalent to
q/e=1.1x1023 electrons (e=1.6x10–19 C).
Example: Suppose the electric potential difference between the inside and the outside of
a living cell wall is 95 mV, with the outside being more positive. In order to maintain the
internal electrical balance, the cell pumps out sodium ions. If there is a hole in the cell
wall which allows a sodium ion to leak in freely, how fast will the ion be moving as soon
as it enters the cell? The atomic mass of a sodium ion is 23 g/mole.
Solution: The ion drops down in electric potential energy by e·∆V=95 meV, which it
gains back in kinetic energy. Since mv2/2=95 meV,
Example: Suppose 4 charges, q1, q2, q3, and q4, each of mass m, are stuck at the 4
vertices of a square of side L. If q4 suddenly gets unstuck and flies off, what is the
maximum velocity that q2 will attain far away from the square?
Solution: The work (energy) required to bring q4 from infinity to the empty vertex is
converted to kinetic energy when the charge flies off. Since the electric potential at
vertex 4 due to the other 3 charges is V=(k/L)·(q1+q2+q3), the work W=q4·V.
Therefore, mv2/2=W or v=(2·W/m)1/2=[2q4·(k/L)·(q1+q2+q3)/m)1/2.
5 µC 6 µC
Example: How much work is required to assemble the triangular array of charges shown
starting from three charges initially located infinitely far away from each other?
Solution: We can calculated the energy by bringing one charge at a time from infinitely
far away to assemble the triangle. The first charge, say q1=2 C, costs no energy. The
second charge, say q2=5 C, costs kq1·q2/3=.03 J. The third charge q3=6 C, which is
affected by the first two charges, costs k·q2·q3/4+k·q1·q3/5=0.0891 J. The three
pairwise interactions add up to 0.119 J. You should convince yourself that the order for
bringing in the charges does not affect the answer.
A Message from Heaven
People have a long history of trying to influence the weather—mostly, in vain. The most
familiar example is probably the attempt of the early Indians to produce rainfall with
certain dancing rituals. To date, very little objective evidence exists which suggests that
weather can be altered, in a controlled fashion (without disastrous consequences), by
human craftiness. Seeding the clouds in order to encourage precipitation appears to be
one of the few (moderately) effective tricks of the trade.
Unaware of the futility of messing with Mother Nature, many people in the 1700s
believed that they could actually prevent lightning. Perhaps they reasoned that if faith can
move mountains, then the fluffy thunderclouds should be no match for the strong of faith.
They may also have reasoned that the effect of their faith might be reinforced by the
intervention of certain would-be naturalists. Accordingly, it was common practice in
those days for people to climb up to the church steeples during stormy weather and ring
the bells as loudly as possible. Evidently, these people believed that sound, which clearly
causes solid objects to vibrate, may help break up the clouds.
Unfortunately for many of these people, their faith turned out to be not only
insufficient but also misplaced. Since the metallic bell was grounded, just like a lightning
rod, many bellringers were struck down by lightning. (Is there a message from Heaven
here about presumptuousness?) In spite of its casualties and ineffectiveness, the practice
continued for many years, perhaps because the clouds eventually and inevitably dispersed
on their own. Ironically, when the practice was finally outlawed, excessive noise rather
than high risk was cited as the primary consideration.
The Burning Bush
We have all heard, at one time or another, the story of Moses and the burning bush. But
was this miracle really an instance of divine intervention or simply a rarity of nature?
The following explanation supports the latter, perhaps more mundane, possibility.
During stormy weather, the potential difference between the Earth and the atmosphere
can be so great that a sudden luminous discharge in the form of lightning can take place.
Under somewhat milder conditions, the electric field in the atmosphere can be just high
enough to cause electrons to escape a charged object more slowly, ionize the air around
the object, and cause an eery glow. When this phenomenon occurs around high-voltage
power lines, it is called a corona discharge and is often accompanied by a power loss.
When the glow is seen around pointed objects, it is often called St. Elmo's fire, named
after the patron saint of sailors.
Historically, St. Elmo's fire has been seen around masts of ships, which may have
been charged by friction with water. In more modern times, airplanes flying through rain
or snow may exhibit a similar glow around the tips of the wings. Perhaps with certain
religious overtones, the glow is often seen also around church steeples, and yes,
occasionally even around bushes.
Example: Elmo through Ermo is a corruption of St. Erasmus, a 4th century Syrian
bishop, who came to be regarded as the patron Saint of sailors. According to legend, just
before St. Elmo died in a storm at sea, he promised to return in some form to let sailors
facing a shipwreck know that they would survive. Sailors often believed that two or more
flames signified good luck and that they were protected by the gods. A single flame
meant that the worst was yet to come. Is there any physical basis for this belief?
Solution: Most of the time, St. Elmo’s fire is observed toward the end of a thunderstorm,
when the air is only slightly charged. From this point of view, multiple occurrences of St.
Elmo’s fire may indeed suggest good luck.
The Hindenburg Disaster
As mentioned earlier, the Hindenburg was a German-built zeppelin which, like the
Titanic, is now remembered mostly for the disaster rather than for the remarkable
engineering. So what exactly caused the disaster? Perhaps the most important factor was
the use of hydrogen—a dangerously flammable gas—for buoyancy. Although it was
ultimately the hydrogen that fueled the explosion, the spark that ignited it was
inadvertently caused by another poor choice of material.
As the Hindenburg was ready to land, handling ropes were dropped to the ground
crew about 43 m below. Due to the electrical storm, both the atmosphere and the
zeppelin became charged. As the ropes became wet, they formed a conducting path to the
ground. As a result, the metal framework, to which the ropes were connected, became
electrically grounded (i.e., had the same potential as the ground). Ordinarily, this should
have also grounded the outer fabric which covered the framework. Unfortunately,
however, the Hindenburg was the first (and the last) zeppelin to have its fabric painted
with a sealant of very low electrical conductivity. As an insulator, the fabric remained
charged to a very high electrical potential, even though the framework had already been
As fate would have it, the handling of the ropes apparently ruptured one of the
hydrogen cells. The released hydrogen presumably caused the reported rippling of the
outer fabric, about one-third of the way from the stern. Seconds after the rippling, a
flame erupted from the same region. Evidently, the large potential difference between the
charged outer fabric and the grounded framework produced a spark across the hydrogen-
filled region. The rest became an inevitable chain reaction of destruction, which ended
with 36 people dead and many more injured.
[The Hindenburg explosion over Lakehurst Naval Air Station in 1937.]
Bovine Distress Syndrome*
During an electrical storm, it is quite safe to shelter either in houses or within an angle of
45º formed by telephone and power lines. It is not safe, however, to shelter under trees
since a spark can jump down from one of the lower branches. Many uninformed cows
have been killed in an attempt to take shelter under trees during an electrical storm. If a
tree is charged to a potential difference of 106 volts and a cow faces the tree, the cow may
feel a voltage of several hundred thousand volts across its body—enough to cause
substantial bovine distress. Since the potential difference decreases radially outward
from the tree (thus, equipotential lines are circles concentric with the tree), the smarter
cows tend to stand sideways to the tree and avoid the unpleasant experience of a large
Similar considerations apply near a fallen high-voltage power line. Although
most people probably know better than to touch the power line, they probably don’t
realize the dangers of simply approaching one. The high potential difference between the
line and the ground produces an intense electric current, which radiates out from the
power line and disperses into the ground around it. Since the current is most concentrated
near the power line, a safe distance must always be kept. In the hopefully infrequent
event that a power line falls down close to you, your best strategy would be to keep your
feet close together, so as to avoid a large potential drop across your body, and hop away
A Shocking Filling
A common filling for tooth cavities is ―dental amalgam,‖ a solid solution made by
dissolving tin and silver in mercury. Although amalgam fillings are inert and generally
do not cause health problems, they can lead to quite a surprise if one bites on a piece of
aluminum foil. The electrochemical reactions between the aluminum and the components
of an amalgam, with saliva and gum tissue acting as the electrolyte, can shock the tooth’s
nerve by sending a small flow of electrons to it. Some of the half reactions involved, with
standard reduction potentials at 25ºC, are shown in the table below.
Al +3 (aq) + 3e —> Al (s) -1.66
Sn+2 (aq) + 3Ag (s) + 2e —> Ag3Sn (s) -0.05
3Hg2+2 (aq) + 4Ag (s) + 6e —> 2Ag2Hg3 (s) +0.85
What happens when you sandwich a strip of cardboard soaked with salt water between
two metallic plates, one of silver and the other of zinc? Admittedly, your list of everyday
activities is not likely to include this particular one. However, if it does, then you'll find
that you have reinvented Alessandro Volta's original electrochemical cell of 1800. If you
connect the two plates by a copper wire, as did Volta, you should find that a current is
generated across the wire. The mechanism relates to the different electron affinities of
the two metals.
In chemistry, the transfer of electrons from one element (or compound) to another
is called a redox reaction. The reactant which donates or loses its electron(s) is said to
be oxidized, while its partner reactant is said to be reduced (hence the name redox).
Table[reduction potentials] shows the different tendencies of several elements to be
reduced. Roughly speaking, the more positive the reduction potential, the greater the
electron affinity of a given element. In Volta's sandwich, for instance, silver is reduced
by zinc. Conversely, zinc is oxidized by silver. The net effect will be a transfer of
electrons from zinc to silver.
cardboard + NaCl
The function of the briny cardboard in Volta's sandwich is to neutralize the charge
separation that occurs as a result of the redox reaction. If charge were allowed to build up
without the neutralizing salts, then the negatively charged silver would repel the incoming
electrons and the current would cease. When functioning as a source of electricity, the
electrochemical cell is usually referred to as a battery, and the two terminals of the wire
are known as the electrodes.1 The term battery really refers to a battery or series of cells,
as is generally the case for typical batteries. The ordinary 12-V car battery, for instance,
is composed of 6 2-V cells, connected in series, which use plates of lead and lead oxide in
sulfuric acid. The ionic solution, or occasionally a solid, acts as the neutralizing medium
and is called the electrolyte. In general, the greater the difference in reduction potential
(or electron affinity), the greater the voltage of the cell. However, the voltage may also
depend on the electrolyte.
Element Reaction Standard Reduction
Potassium K+ + e– = K –2.925
Sodium Na+ + e– = Na –2.87
Zinc Zn+2 + 2e– = Zn –0.763
Lead Pb+2 + 2e– = Pb –0.126
Copper Cu+2 + 2e– = Cu +0.337
Silver Ag+ + e– = Ag +0.7991
Chlorine Cl2 + 2e– = 2Cl– +1.3595
Gold Au+ + e– = Au +1.68
Seeing Field Lines—Demo
The shapes of field lines for different geometrical arrangements of charges can be
obtained in a variety of ways. One way is to suspend grass seeds in an insulating liquid,
such as oil or glycerine. The electric field polarizes the seeds, which then orient
themselves along the field lines. The field thus becomes visible.
A Lie Worth a Thousand Ohms
A polygraph is an instrument that records changes in such physiological processes as
heartbeat, blood pressure and respiration. In criminology, this device can also be used as
a lie detector. Electrodes are attached a short distance apart on a person's skin and
measure the current due to a small voltage difference between the two electrodes. The lie
detector operates on the principle that most of us experience a certain level of discomfort
1The positive terminal is called the anode, while the negative one the cathode. Referring
to the path of electrons, the names derive from the Greek words for "up road" and "down
when consciously speaking untruthfully. The physiological evidence of untruthfulness is
an increase in perspiration. Since moisture tends to lower the skin's resistance (to the
flow of charge), a greater current is registered between the electrodes. In fact, the
resistance of about 1 MΩ for dry human skin may drop to about 1 kΩ after becoming
covered with a layer of salty water (i.e., sweat). Of course, the smaller the distance
between the electrodes, the smaller the resistance. An expert, asking a series of probing
questions, is always present to ensure that nervousness is not misinterpreted as lying.
Incidentally, people who manage to "fool" the machine are often afflicted with personality
disorders (e.g., psychosis) somewhat more severe than just the occasional inclination to
Many aquatic animals are able to detect the minute electric fields created by ocean
currents moving through the Earth’s magnetic field. These fields help the animals orient
themselves and navigate across the waters. Other groups of fish, particularly from turbid
habitats in Africa and South America, generate their own electric fields by using specially
modified muscle tissue. Such fish can hunt, navigate around obstacles, and communicate
with one another by detecting changes in these electric fields. The electroreceptive cells
that make this sense possible are known as the ampullae of Lorenzini. These cells are
evolutionarily related to hair cells and are concentrated in a series of canals and pits
around the animal’s snout.
In addition to its usefulness in navigation, electrodetection is quite valuable for
certain salt-water predators because essential life processes of potential prey generate
small but detectable electric currents. The exchange of ions across the gills of fishes as
they breathe, for instance, generate electric currents, as do the movements of their
respiratory muscles. Sharks, in particular, are so sensitive to such currents that they can
even detect prey that is completely buried in the sand.
Sharks are possibly the world’s most adept predators. Even the forces of
evolution have been unable to improve on the efficiency with which the shark preys upon
the other sea creatures. So strong is their predatory instinct that sharks often attack even
inanimate objects. Sharks have been found to have cans, tires, and other assorted
goodies, presumably of little nutritional value, lodged in their bellies. With their
voracious appetite, deep-water sharks have even a significant threat to long-distance,
undersea cables. Apparently, the sharks are attracted to the small electric currents
generated by amplifiers positioned at regular intervals along the cables. Intelligence is
probably not their greatest asset.
Because electroreception is only effective over a relatively short range, sharks
must also rely on their other senses, such as an exquisitely sensitive sense of smell. A
bleeding wound will often attract sharks from miles away. And even if the bleeding is
not a sufficiently strong invitation for dinner, sharks are always tuned in to the various
sounds produced in the water. They are particularly sensitive to low-frequency sounds,
around 40 Hz. Conveniently, this happens to be the frequency of sound emitted by many
wounded fish. What makes sharks different from other creatures similarly equipped with
supersensitive detection systems? Jaws!
Question: Various experiments have been performed on sharks in order to measure the
sensitivity of the sharks to various stimuli. Consider the following four different
scenarios: a small, live fish is buried, completely out of sight, in the sand; the fish is
hidden from view by an agar chamber; the agar chamber is now covered by a metalic
sheet; active electrodes, buried in the sand, generate small, random currents. Can you
explain why the shark fails to show interest only in the second case (agar and metal)?
Answer: The sand only blocks visual cues. Agar only blocks scent and visual cues. The
metal casing, on top of the agar, also blocks (shields) electrical cues. In the absence of
visual, scent, and electrical cues, the shark is unable to detect the fish. Note that the shark
mistakes the electrodes for a breathing fish.
[(a) A shark easily finds live fish buried out of sight by homing in on the tiny elecrical impulses produced
by the prey's breathing movements. (b) The shark detects just as easily a fish covered with agar, which
blocks scent cues but not electric currents. (c) An agar chamber covered with an electrically insulating film
successfully hides the fish. (d) A shark dives for electrodes that simulate the electric field of a living fish in
the absence of any scent cues.]
Frozen to the Wire
We are often cautioned by professionals about attempting to reproduce certain potentially
dangerous activities without the benefit of expertise. Benjamin Franklin's famous kite
experiment should serve as a historical example. The first two men who tried to repeat
the experiment were electrocuted. Incidently, even Franklin’s kite was not struck by
lightning, as is commonly believed. Franklin was quite fortunate in this regard. He
simply showed that hairs on the kite string stood apart and suggested that the kite
collected charge from the atmosphere.
In the U.S., accidental electrocutions occur at an alarmingly high rate of about
1000 per year. But just how much electricity does it take to get electrocuted? Roughly
speaking, the effects of current through the human body are as follows:
Effects of Current through Human Body
Current (A) Effect
<0.01 tingling or imperceptible
0.01–0.02 painful and cannot let go (muscle spasms)
0.03 breathing disturbed
0.07 breathing very difficult (could be fatal if
0.10 death due to fibrillation
>0.20 no fibrillation, but severe burning and no
Strangely enough, larger currents can often be far less dangerous than smaller
ones. The intermediate range of 0.1-0.2 amps is ordinarily the most lethal because this
level of current initiates fibrillation, an uncontrolled spastic twitching of the heart. The
resulting disruption of blood flow quickly results in death. Another (controlled) electrical
shock is the usual way to stop fibrillation. Currents significantly above 0.2 A stop the
heart completely. However, the heart often resumes beating normally as soon as the
current ceases. In addition, normal first aid procedure can usually restart it.
The current passing through the victim is usually determined by the skin
resistance, which ranges from about 1 kΩ for wet skin to 0.5 MΩ for dry skin, with the
internal resistance being in the range 100-500 Ω. Touching voltages higher than about
240 V usually results in current puncturing the skin. Often, a person grabs a wire that has
sufficient current (0.01–0.02) to contract his hand muscles onto the wire. That level is
initially not lethal but does become lethal with time as the person’s resistance gradually
drops as a result of sweating. Electricians working with live or potentially live wires
often use the back of their hands or fingers to move the wires, so as not to become
―frozen‖ to the wire.
Question: Is it correct to say that an electrician is safe from shock if he uses the back of
his hand when they touch live wires?
Answer: No. He may still get a shock, but he will not get frozen to the wire because his
hand would close ―away from‖ the wire.
Weapons of the Gods*
Lightning is perhaps nature’s most familiar and most spectacular display of
electricity. Since ancient times, lightning has both awed and fascinated people with its
splendor and might. The early Greeks, for instance, associated the lightning bolt with
Zeus, their most powerful god. In the Bible, lightning is often associated with the wrath
of God. Today, we understand that lightning occurs as a result of charge separation
within thunderclouds. Water droplets within the clouds break up into positively and
negatively charged parts, with the negative ones being heavier. Because of the weight
difference between the two types of particle, the top becomes positively charged, while
the bottom of the cloud becomes negatively charged. The negative charge induces a
positive charge on the surface of the Earth, and a potential difference develops between
the cloud and the ground. If the potential difference is great enough, air becomes ionized
and an electrical path opens up from the cloud to the ground.
A typical lightning bolt delivers to the ground only about 25 coulombs of
(negative) charge, an amount easily obtained from an ordinary car battery. So what is the
source of the enormous power? Simply put, the very high voltage between the cloud and
the ground. The lightning bolt transfers this charge through a potential difference of
typically 10-100 million volts in only 5 milliseconds. This amounts to as much as 1
billion joules of energy and a staggering 200 billion watts of power, enough to supply a
typical home for a few months. The lightning also generates temperatures five times
hotter than the 5,800 ºC found on the surface of the Sun.
Usually, a bolt of lightning consists of several quick strokes, rather than just one
discharge. Before the main thunderflash, a leader works its way down, jumping from one
raindrop to another and ionizing the air in the process. The main stroke then runs down
this path to the ground. If the wind is blowing, the second and third strokes do not follow
the original path exactly.
At any given moment, several thousand thunderstorms are raging on Earth.
Collectively, these thunderstorms produce about 100 lightning bolts every second.2 Only
about 20% of these lightning bolts actually hit the ground, while the rest occur between or
within clouds. Nevertheless, in the U.S. alone, lightning is responsible for over 100
deaths each year.
Example: Estimate the electrical energy and power delivered to earth by the typical
Solution: Assuming the lightning bolt transfers 25 C through a potential difference of
100 million volts, we get E=Q·V=25·108=2.5x109 J. Actually, since the clouds are
discharging, the voltage is not constant but drops to zero at the end of the discharge. It is
therefore more correct to use the average value of (100+0)/2=50 million volts, which
would give an energy of about 1 billion volts. Since the charge transfer last about 25 ms,
we calculate the power from P=E/t=109/0.005=2x1011 watts.
Question: There is an old wives’ tale claiming that lightning tends to seek out oak trees.
One might argue that ―A tree is a tree. How could lightning possibly know the
difference?‖ Amazingly, to the credit of the old wives, a strikingly high proportion of
trees struck and shattered by lightning are, in fact, oak trees. But what can possibly cause
such preferential mistreatment by Mother Nature?
Answer: Of course, every lightning strike does not result in an explosion. If a tree
happens to be thoroughly wet, from top to bottom, the current simply descends through
the outer water layer and leaves the tree unharmed. If not, the current may enter the tree
at the top and descend through the sap. The rapid heating and expansion of the sap can
then blow the tree apart. Oaks are more susceptible to explosion because their rough
barks can make it difficult for the rain to wet their bottoms, so to speak. By contrast,
smooth-barked trees become wet quickly and completely, as rain trickles down relatively
unimpeded along their bark.
Question: Why do gushes of rain or hail often closely follow lightning strokes during
Answer: Normally, charged or polarized water droplets in the clouds are partially
supported by local electric fields within the clouds. However, right after the cloud
discharges through a lightning stroke, the internal electric fields diminish and some of the
water droplets precipitate.
2E. R. Williams, “The Electrification of Thunderstorms” Scientific American, Nov 1988
The flow of electrical charge through some material is called the current, which is
formally defined as
[electric current] I=Q/t
where Q is the amount of charge crossing some surface in a time t. Of course, bigger
surfaces allow more current to pass through under the same conditions. Voltage turns out
to be the driving force for current flow, which should not be very surprising since both V
and I arise from a separation of charge. Since for every action (force) there must be a
reaction (opposite force), we come to
[Ohm's law] V=IR
where the resistance R describes the opposition to current flow. Ohm's law simply says
that the greater the driving force (V), the greater the current. On the other hand, for a
given V, the greater the resistance, the smaller the current. As far as units go, V is in
volts (V), I is in amperes (A), and R is in ohms (Ω).
Example: A flashlight uses 3 volts to drive a current through a thin wire of resistance R
to produce a current of 0.4 A in the wire. The wire heats up from the frictional flow of
charge through it and produces light. What is the resistance R of the wire?
Solution: R=V/I=3/0.4=7.5 Ω.
The Power Touch
The resistance of a wire changes when it is strained (stretched). By monitoring the
current through a fine metal wire mounted alongside a bridge, machines can indicate
strain on the bridge based on the wire’s varying resistance. Such a device is called a
strain gauge, and can be used to equip robots with the sense of touch. One of the most
promising sensor designs consists of an electronic chip printed with a fine metallic
network and covered by a thin sheet of conducting rubber. Pressure on the rubber
changes the resistance of the chip, transmitting a ―picture‖ of the strain on the robotic
skin. As recently as 1985, robotic hands, possessing superhuman strength, had the
dexterity to gently crack an egg into a mixing bowl.
Because electricity costs money, we often need to calculate the electric power associated
with any current flow. Power is simply the average energy per unit time or E/t=qV/t=IV.
Alternatively, since V=I·R, power can be expressed in two other, equivalent forms:
[electric power] P=IV=V2/R=I2R
Example: A cigarette lighter in a car is a resistor that, when activated, is connected
across the 12-V battery. If the lighter dissipates 33 W of power, find the resistance in the
lighter and the current that it draws from the battery.
Solution: R=V2/P=122/33=4.36 Ω and I=P/V=33/12=2.75 A.
r r r r
r r r infinite
r r r r
r R R
Example: Suppose a voltage difference ∆V is applied across the terminals of the infinite
resistor network shown in the figure. What is the rate of heat production (power)?
Solution: First, we need to find the effective resistance R of the whole network. Imagine
cutting the array into two parts, as shown, such that the left side includes 3 resistors and
the right side looks the same as the original. (Since the array in infinite, removing a small
section from it should not affect it.) Two of the r's (the ones on the ends) are in series,
while the third r is in parallel with R. Therefore, the combination is equivalent to
2r+(1/r+1/R)–1=R. Using the quadratic equation and eliminating the negative resistance,
we get R=(1+√3)·r. The power is then just P=(∆V)2/R.
Question: How does a high voltage allow for economical transmission of electric
Solution: In order to minimize resistive heating in the power lines, the current must be
minimized. For a given quantity of power P=I·V, a high voltage means a low current.
Question: Power companies try to minimize the current in long-distance transmission
lines in order to avoid generating excessive heat. Typically, they step up the voltage to
120,000 volts. Since even lower currents would be produced by higher voltages, what
prevents the company from using millions of volts instead?
Solution: Too high a voltage difference between two power lines or between a power
line and the ground would cause an arc (i.e., dielectric breakdown of air) similar to a
The Hour of Power
Most modern appliances consume many kilowatts of energy per day. A 100-watt light
bulb, for instance, consumes 100 J in one second or 100·3600=360,000 J in one hour. In
order to avoid dealing with such huge numbers and scaring the customers, electric
companies prefer to use an energy unit called a kilowatt·hour (kWh). One kWh is equal
to 1000·3600=3.6x106 J, the energy consumed by a 1-kW device in one hour.
The resistance R not only depends on the intrinsic properties of the material, but
also on the geometry involved. A large cross-section A for the current to flow through
and a short distance L over which to flow both contribute to a smaller resistance. This
where , called the resistivity, is only a property of the material.
Example: Compare the resistance of two identical wires side by side and end to end.
Solution: Side by side, A doubles and the R falls by a factor of 2 compared with one
wire. End to end, the two wires effectively have twice the length of one wire and R
increases by 2. Thus, end-to-end resistance is 4 times greater than side-by-side resistance.
The last example introduces the concepts of resistors in series (end to end) and in
parallel (side by side). In general, in-series resistors feel the same current and divide the
voltage, while in-parallel resistors feel the same voltage and divide the current. As far as
the net current and voltage difference between two points is concerned, any complex
arrangement of resistors can be replaced by one effective resistor. The recipe for doing
this involves replacing in-series combinations with
[in series] Rs=R1+R2+
and in-parallel combinations with
[in parallel] 1/Rp=1/R1+1/R2+
Example: Suppose a Christmas tree is decorated with a long wire of 100 light bulbs,
each of which has a filament with a resistance of 10 . You notice that whenever you
remove any one light bulb, none of the others seem to work. If the voltage source for the
lights is simply the wall outlet, how much current do the light bulbs draw from the wall?
Solution: Since one light bulb breaks the circuit for the rest, all the light bulbs must be in
series. In other words, they all feel the same current. Since the net effective resistance is
Rs=100·10=1000 Ω, the current is 120/1000=0.12 A.
Example: Imagine a cube, with each edge carrying a resistor R. If you apply a voltage V
between any two corners furthest away from each other, what is the current between the
two corners? Assume that current can only flow through these resistors.
Solution: The current splits as shown in the figure. If we pick any path connecting the
two terminals we get V=(I/3)·R+(I/6)R+(I/3)·R=I·(5R/6). Thus, the effective resistance
is Reff=5R/6 and the current is I=V/Reff=6V/5R.
Example: A 500-watt toaster, a 900-watt microwave, and a 100-watt lamp all operate on
the same circuit. When a 300-watt coffeemaker is connected to the same circuit, the fuse
melts. What is the amp rating of the fuse?
Solution: Each device draws some current from a common voltage source (V=120V).
Thus, 500+900+100<ImaxV<500+900+100+300 or 12.5<Imax<15.
These rules, which are almost pure common sense, help us figure out the currents that
flow in any circuit, no matter how complicated. The junction rule says that the net
current entering a junction (i.e., a point where several wires meet) must equal the net
current leaving the junction. The reason is simply that charge cannot accumulate in any
point along a wire. The loop rule says that around any closed circuit loop, the sum of the
potential rises must equal the sum of the potential drops. This rule is similar to the
conservation of gravitational potential energy: no matter how many times you go up and
down, your gravitational potential energy will not change if you end up at the same
Example: Find the currents in all three resistors shown in the figure and the power
dissipated in the 1-Ω resistor.
Solution: The bottom loop gives 4–3·I2+2·I1–1=0 and the top loop gives 1–2·I1+1·I=0.
The big (outside) loop does not provide any new information. The lower two currents
merge to form I=I1+I2. Thus, we have three unknowns and three equations, which can be
solved (with a little bit of luck) to obtain I1=0 A, I2=1 A, I=1 A. The power dissipated in
the 1-Ω resistor is I2·R=12·1=1 W.
Most modern automobiles are equipped with windshield wipers that can operate
intermittently, as during a light rain. The wipers are part of an RC circuit whose time
constant can be varied by selecting different values of R through a multi-positioned
switch. A small R allows a large current to charge the capacitor very quickly.
In a typical defibrillator, a 16 F capacitor is charged to about 6000 volts and is then
discharged in about 2 ms. The energy is about 300 J (=QV/2), the charge 96 mC (=CV),
while the current is about 50 A (=Q/t). Such a current is certainly enough to stop all heart
action. On the other hand, when the heart has already stopped, this current can evidently
get it to beat again.
Question: A defibrillator is a device used by emergency medical teams to stop the heart
fibrillation (uncontrolled twitching) of a heart attack victim. In the portable version, a
battery charges a capacitor to a high potential difference (around 5 kV) in somewhat less
than a minute. Conducting leads called ―paddles‖ are placed on the victim’s chest and
then allowed to discharge through the victim. Assuming the capacitance is 70 µF, the
stored energy must be U=C·V2/2=875 J. About 200 J of this energy is sent through the
victim during a pulse lasting about 2 ms. The resulting power is 200/0.002=100 kW,
much greater than the power used in the charging process.
The Casual Electrician
There is an old saying by people who work with high voltages: ―Keep one hand in your
pocket.‖ Is this the hip version of ―keep both hands on the wheel?‖ Whether one keeps
his hand in the pocket or simply wears a glove, the second hand must be electrically
isolated from any grounded object. If, for instance, the left hand touched a metal chair
while the right hand was in contanct with high voltage, a conducting path would be
created between the two hands. The result would be much worse than any heartburn.
Spare the Rod and Spoil the House
The lightning rod was one of the many significant contributions to mankind made by the
great American scientist and statesman, Benjamin Franklin. Based on his understanding
of the *** principle, Franklin argued that the tip of a lightning rod should be as sharp as
possible. After all, a sharper metallic tip would attract electric charges better, therefore
enabling the lightning rod to channel away as much as possible the potentially disastrous
charges released in a lightning. The lightning rod is welded to a heavy conducting cable
which runs down into the ground. When lightning strikes, the current is conducted
through the cable rather than through the house.
Most of the lightning rods used in the U.S. had sharp tips, including the one at the
top of the towering Empire State Building in New York City, which was the world's
tallest building (and consequently the most likely target of lightning strike in a
thunderstorm over Manhattan).
Meanwhile in Britain, King George III had some ideas of his own. Out of whatever
argument, he somehow believed that a spherical tip would be better suited than a sharp
one for the lightning rod. Exercising his royal power, he forcefully ordered all the
lightning rods in Britain to be made with a spherical tip, including those that protected the
Well, guess who was right? No, it was ....... the king!!! Believe it or not, Ben Franklin,
the great American scientist, actually lost this debate to a mere British royalty who
supposedly knew little about the sophisticated science called electricity. It turned out that,
while the Empire State Building equipped with sharp-tipped lightning rods suffered from
as many as 23 lightning strikes per year on average, the Buckingham Palace was never
struck even once during the same period of time.
So what was wrong with Franklin's argument? This is not an easy question. After
two decades of exhaustive studies, an American physicist named Charlie Moore***
finally came through with the answer. His experimental results suggested that the electric
field at the tip of a dull-shaped lightning rod was actually 1.5 to 2 times as much as that
for a sharp-tipped one. A spherical tip with a diameter of around 1 in. is probably best at
channeling the lightning current through. Reason? As electric charges accumulate at the
tip of a needle-shaped lightning rod, the extremely strong electric field is actually able to
ionize the air around it, creating an ionized shield around the tip of the lightning rod. This
shield would retard further induction of lightning current through the rod. Just ask the
king about it!
Question: A lightning rod provides a cone of protection whose ground radius
approximately equals its height above the ground. What the heck does that mean?
The Third Prong*
Faulty electrical devices can deliver dangerous, and sometimes lethal, shocks. In the
most common scenario, the high-voltage wire of an electric device is shorted (i.e.,
becomes connected) to the case due to frayed insulation on the wire. If a person who is
grounded touches the case, he will complete a path for the current to the ground. Since a
person’s resistance is typically much smaller than the resistance within the device, the
person presents the path of least resistance and will therefore draw a large current. This
kind of hazard can be avoided by the use of a three-wire system. The third prong of a
plug is connected by a wire to the case of the appliance, while the third hole in the outlet
is connected to the ground. In the event of faulty wiring in the appliance, the current is
routed to the ground through the prong and not through the person because the case and
the ground have essentially no resistance between them.
Example: How much current will a person who presents a resistance of 100,000 Ω draw
from a 10,000-V device?
Solution: Using Ohm’s law, we get I=V/R=0.1 A, a lethal dosage.
The electric eel (Electrophorus electricus), found in South American rivers, sometimes
exceeds six feet in length. However, it uses more than its size to intimidate its prey. A
healthy eel can generate lethal currents as large as 1 ampere, arising from a potential
difference of several hundred volts from head to tail. The eel has specialized cells called
electroplaques in its body that produce a flow of ions (i.e., current) across their
membranes when triggered by a signal from the brain. The membrane of each
electroplaque has an emf (voltage) of 0.15 V and an internal resistance of 0.25 .
Several thousand of these cells are arranged in series, which are then grouped in several
hundred (or more) parallel combinations. The series connections ensure a large enough
voltage, while the parallel arrangement maintains a high enough current to stun or kill the
The numerical examples given below serve to illustrate these points. Basically,
the resistance of the water is generally much greater than the internal resistance of any
one of the million or so electroplaques. If all the electroplaques were arranged in series,
the current through the series would be large enough to damage the eel itself. An all-
parallel arrangement would cause the eel to be too wide and would thus hinder the eel’s
mobility. The trick is to use a combination of series and parallel ―wiring‖. In this case,
the current through each cell is kept low without sacrificing the external current because
many little currents inside the eel add up to produce one large current in the water. In
practice, the distribution of the electroplaques differs somewhat from species to species,
but the basic ―wiring‖ remains the same.3
Example: Suppose the resistance Rw of the water is around 1000 . Calculate the
current that would be generated by the eel if the electroplaques were all arranged in
series. How about in parallel? Assume the eel has 1 million electroplaques.
3H. Grundfest, “Electric Fishes.” Scientific American, 203, 115, March 1960
Solution: In an all-series arrangement, many little voltages would add up, and the current
through the water—and through the eel—would be 0.15x106/(1000+0.25x106)=0.6 A.
On the other hand, in an all-parallel arrangement, many little currents would add up to
produce a net current of 106·0.15/(1000+0.25)=150 A.
Example: Now calculate the current generated by the eel if the cells are arranged in 200
parallel rows, each having 5000 cells in series.
Solution: Each row can be represented by an emf of 750 (=5000*0.15) volts across a
resistance Rrow=1250 . The total effective resistance of the eel is then
1250/200=6.25 , in series with 1000 in the water, all driven by an emf of 750 volts.
The current through the water is therefore 750/(6.25+1000)=0.75 A, while the eel feels
only 0.75/200=0.004 A in each individual row.
Example: Assuming an electroplaque is around 10 microns in diameter, what would the
minimum length of the eel have to be to accommodate a million electroplaques in series?
Solution: If the cells are lined up end to end, the total length will be 106·10x10–6=10
m—a pretty long fish!
Question: Freshwater eels have a greater number of electroplaques in series than do the
eels living in briny water. Can you see why?
Answer: Salt increases the conductivity (i.e., lowers the resistance) of water. Thus,
freshwater eels need more of an in-series voltage component in order to generate the same
current through the higher resistance of the fresh water.
A Bird’s Wingspan*
A common sight, often taken for granted, is that of a bird sitting on a high-voltage cable.
Do birds have some unusual ability to withstand high voltage? In a word, no. Since it is
the current that’s dangerous and not the voltage, a bird sitting peacefully on one cable
does not draw any current through its body. On the other hand, if the bird were to touch
both the high- and the low-voltage cables at the same time, it would then draw a lethal
current through its body. For this reason, power cables are often set up in such a way that
the separation between the high- and the low-voltage cables exceeds the bird’s wingspan.
Certain crystals such as quartz, Rochelle salt, (NH4)H2PO4
(ADP) and LiNbO3, possess an important electro-chemical
property called the piezoelectric effect. Basically, this is
a process which involves the transformation of mechanical
energy into electrical energy, and vice versa. On the one
hand, an electric polarization can be established in these
materials with the application of a mechanical pressure
(strain). In this process, the mechanical work done on the
crystal through the application of the pressure is
transformed into electrical form. This is referred to as
the direct piezoelectric effect. On the other hand, a
mechanical stress can be produced in these materials as a
result of the electric polarization of the crystal. In this
case, electrical energy is transformed into its mechanical
counterpart. This is referred to as the converse
The piezoelectric effect makes it possible to rapidly switch
between mechanical and electrical signals. It has found
many applications in modern electronics, including
telephones, quartz watches, etc. In medical applications,
for instance, an alternating voltage applied to the opposite
faces of a piezoelectric crystal (such as quartz or
strontium titanate) causes the crystal to vibrate at the
same frequency as the voltage. As the crystal vibrates, it
emits a beam of ultrasonic (>20 kHz) waves. Since the
piezoelectric effect is reversible, a single crystal can be
used to both transmit and receive ultrasonic waves.
Piezoelectricity and Autofocusing
In recent years, autofocusing vidoecameras have made their ways into millions of
households. The ways these cameras focus automatically vary. Among the most effective
ones is the passive electro-optical mechanism utilizing piezoelectric effect, such as those
found in many Canon models.
To adjust the camera for the best focus, a motor has to be used to drive the lens forward
or backward until a perfect position can be found. As the microcomputer on-board the
video camera senses the right direction for the motor to drive the lens, it sends out an
electric signal in terms of an electric voltage pulse. This electric signal is applied to a
piece of quartz crystal inside the camera. The resulting electric polarization of the crystal
causes it to produce a precise mechanical strain, which is then used to drive the
mechanical motion of the lens. The piezoelectric effect is basically instantaneous, thus
allowing for swift and continuous autofocusing, even when the object is in rapid motion.
(For more on autofocusing, refer to the section entitled, "A Dummy's Guide to Picture-
Taking", in Optics.)
An Underground Power Plant
From the example above, it is conceivable that if we lay some piezoelectric material
underneath a major highway, it can be polarized and consequently produce electric
charges upon being pressurized by passing vehicles. If there is an effective way to utilize
the resultant electrostatic potential, then we would have succeeded in building an
underground power plant that generates electricity whenever there is some traffic (which,
as we know all too well, is in no short supply here in L.A.)!
The Haunted Railroad
In a small town in New Jersey a few years ago, a section of a railroad was reportedly
haunted—mysterious, ghostly lights, typically spherical, were seen emanating at night.
Many were scared by this seemingly supernatural phenomenon. Other, more
scientifically oriented, individuals suggested that the railroad somehow concentrated
sufficient electric charge to produce the eerie glow. At any rate, by popular demand, a
section of the railroad track, where the ghosts were seen, was dismantled. However, to
the dismay of many people, the ghosts persisted.
Subsequently, a group of scientists conducted a survey of related phenomena and
discovered a number of similar incidents across the U.S. Geological investigation
revealed a feature common to all of these occurrences: a location at or near a geological
fault, with a large quantity of quartz underneath. An interesting puzzle! As usual,
physics provides us with the answer. Quartz happens to be one of the crystals which
exhibits the piezoelectric effect. The pressure, perhaps as much as 3,000 tons per square
meter, was provided by the tectonic movement of the earth near the fault line. The
polarization charge produced by such pressure evidently migrated to the surface of the
earth, where an electric field was then created. When the field was great enough (i.e.,
sufficient surface charge had accumulated), the air near the surface became ionized and a
glowing discharge, similar to St. Elmo’s fire, was produced. Leave it to the physicists to
bust yet another perfectly respectable ghost!
Charges tend to accumulate around sharp points of a conductor. As a result, electric
fields around these points can be quite high and can give rise to a number of interesting
phenomena. The following example presents one particular instance where the electric
field is indeed higher around the more curved sections of a conductor.
Example: Suppose two charged, conducting spheres of radii R1 and R2 are separated by
a long distance and connected by a conducting wire. Find the ratio E1/E2 of the electric
Solution: The potential of a uniformly charged spherical shell of charge Q is given by
V=kQ/R. Since the two spheres are connected by a conductor, they must have the same
potential. Thus, kQ1/R1=kQ2/R2 or Q1/Q2=R1/R2. In other words, the bigger sphere
carries more charge. Since the electric field at the surface of a charged sphere is given by
E=kQ/R2, E1/E2=R2/R1. We see that the electric fields are inversely proportional to size
of the spheres.
The Woodstock of Physics
In 1911, the Dutch physicist Kammerlingh Onnes discovered that the resistivity of
mercury absolutely disappears at temperatures below about 4 K. Materials which have
zero resitance are called superconductors. The impact of superconductivity and its
importance in technology cannot be overstated. Currents induced in a superconducting
ring, for example, have persisted for several years with no apparent decay. A large
superconducting ring is now being used in Tacoma, Washington, to store up to 5 MW of
electrical energy to be released during peaks in demand.
Until 1986, all known superconductors required cooling to extremely low
temperatures, which were generally achieved with liquid helium. However, the expense
(and cumbersome machinery) associated with such cooling precluded an everyday use of
superconductors. In 1986, however, new ceramic materials were discovered that become
superconducting at considerably higher temperatures. Working at the IBM Zurich
Research Laboratory, Johannes G. Bednorz and Karl A. Müller reported evidence for
superconductivity near 35 K in an oxide of barium, lanthanum, and copper. Bednorz and
Müller sparked vigorous research activity in the field of superconductivity which
culminated in 1987 with the discovery of the so-called high-temperature superconductors
which could be cooled with liquid nitrogen. Liquid nitrogen, whose normal boiling
temperature of is 77 K, is cheaper than bottled water and can easily replace the much
more expensive liquid helium.
The excitement over superconductivity can perhaps be epitomized by the so-called
―Woodstock of physics,‖ a 1987 national meeting of over 2,000 physicists, jammed into
an all-night session to hear the latest reports on newly discovered superconducting
materials. For their important contribution to science, Bednorz and Müller were awarded
the Nobel Prize in 1987. Today, there are thousands of known superconductors and
Table[superconductors] lists a few of them. At this point, room-temperature
superconductivity cannot be ruled out.
Critical Temperatures for Various
Material Tc (K)