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Transcript
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Notes follow and parts taken from Physics (6th Edition, Cutnell & Johnson)
Electricity & Charge
All normal matter is made up of charged particles. “Charge” is just a way to measure how
strongly two objects attract (or repel) one another electrically. We’ll see that charge is similar to
mass in that the product of the force between two charges depends on the sizes of those charges
and is inversely proportional to the square of the distance separating them, just as we saw when
we studied the law of gravity. Notable differences between the two situations include the
possibility of repulsion between like charges (all masses attract one another gravitationally –
even antimatter) and the vastly greater strength of the electric force.
We observe charges to be quantized (i.e., not continuous, but having a smallest possible size,
just as we now know water is not a continuous substance but a large number of molecules) in
units of the charge on the electron, usually represented as e. The SI unit of charge is called the
Coulomb (C) (we’ll define it later) and it is a very large number of charges. In fact, each
electron has a charge of only -1.6 x 10-19 C. That means it would take a few billion billion of
them to make one Coulomb. By convention, the charge on the electron is defined as negative,
meaning the charge on the proton is positive. The signs are opposite, but the charges are exactly
the same magnitude, to the highest accuracy we can measure.
As always, there will be a few enhancements and corrections to the statements above that we’ll
examine later. One is that experiments tell us that we can actually have charges which are 1/3
and 2/3 the size of the basic charge e when we look at the particles which make up protons and
neutrons (called quarks). These have never been isolated individually, but their existence has
been demonstrated indirectly. For this reason, we’ll still consider e to be the fundamental unit of
charge.
You’ve probably known since childhood that like charges repel and unlike charges attract. To
this idea we’ll add the fact that charge is conserved. This law is absolute. No experiment has
ever shown the overall destruction or creation of charge. We can separate charges very easily –
rubbing a glass rod with a piece of fur, or your hair with a balloon will demonstrate this. These
are methods of mechanically transferring electrons from one place to another. When you do this,
the material that gave up the electrons becomes positively charged (having lost some amount of
negative charge) and the material that picks up the electrons is negatively charged. The sum of
the charges, though, is still zero (assuming the objects were neutral in the first place).
Conductors & Insulators
Just as we found that certain materials conduct heat better than others, certain materials conduct
electricity better than others (usually, we see that good conductors of heat are also good
conductors of electricity). In a good conductor, the outermost electrons of the atoms (at least
some of them) are not tightly bound. They are essentially free to roam around the conductor, and
when they move, they carry charge from one place to another (generating a current, which we’ll
talk much more about in a few chapters), or conduct electricity. Among the best conductors are
gold, silver, copper, and aluminum (they’re also very shiny, and that’s not a coincidence).
Things like quartz, glass, rubber, most plastics, wood, etc. are not good conductors, and are
therefore called insulators. The atoms in these materials hold their electrons very tightly.
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Charging Materials
If we put an excess of electrons on an object, we say that it has been charged. We could also take
electrons from the object, and put a positive charge on it. If we want to try this on a conducting
sphere, it needs to be insulated from the Earth. The reason is that the Earth is a huge conductor,
and will gladly grab all of the excess charge it can reach. This is why we use lightning rods to
direct lightning to the Earth – it can soak up the charge easily.
We can charge the object by touching it with a charged rod and letting some of the charge
transfer (it’s easier to say that than to say “letting electrons move from the object to the rod if the
rod is positively charged” or “letting electrons move to the object from the rod if the rod is
negatively charged”). The excess charge will quickly (very quickly) move to the surface of the
metal sphere. It will do this because that’s the way it can get as far away from the other charges
of the same sign that came over with it from the rod. This is charging by contact.
We can also charge by induction. Imagine connecting the metal sphere to the Earth
(“grounding” it) and bringing a negatively charged rod nearby. The electrons in the sphere near
the rod are pushed away, and the grounding wire gives them an exit to use. Some of them leave
the sphere entirely, and will only return when the charged rod leaves. What if we break the
grounding wire connection before letting the rod leave? Now we’ve driven electrons into the
Earth, and they can’t get back to the sphere, so it is left with an overall negative charge. (Notice
that it’s exactly the same size as the opposite charge we’ve added to the Earth, so we’re still not
creating or destroying charge, just moving it).
We couldn’t do the same trick with an insulating sphere, because the electrons near the rod aren’t
free to move large distances away from their atoms, much less leave entirely. All they can do is
shift position a small amount (small on the atomic scale) and spend more time on the far side of
the atom they’re bound to. That will tend to make the insulating sphere’s surface slightly
positive and give a small attraction between it and the rod.
Coulomb’s Law
Now that we know there is a force between all charges, we should find out the form of it. We’ll
find that it looks almost exactly like Newton’s law of gravity, except we’ll replace mass with
charge, since it’s generating the attractive force. As in the case of gravity, we’ll multiply by the
inverse square of the distance between the two things, and we’ll also need a constant to make the
units balance. This one is sometimes written as k and its value in the SI system is 8.99 x 109 N
m2/C2 (notice the similarity in the units between this and “G” – we replace kg by C, but that’s it).
Notice that while G is small, k is very large. This reflects the difference in strengths of the two
forces. For reasons we’ll look at later, we sometimes write k in a different form: k = 1 / ( 4 π ε 0 )
where ε0 is called the permittivity of free space and equals 8.85 x 10-12 C2 / (N m2). Therefore,
we write Coulomb’s law as
F=
k q1 q 2
q1 q 2
=
r2
4π ε 0 r 2
where q1 and q2 are the sizes of the two charges. If they are the same sign, we’ll get a positive
force (repulsion). If they’re opposite, we’ll get a negative force (attraction).
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This force is, like all forces, a vector. That means it has magnitude and direction. The direction
will be either towards the other charge (attraction) or away from it (repulsion). In one sense, now
that we know the form for the force, our work with Coulomb’s law is done. It’s now just one
more force we need to consider when determining the motion of an object. It’s part of Newton’s
2nd law as a force to be included in
∑F
=ma
in the x direction, with a similar force in
x
x
the y and z directions. We’ll find its components just as we’ve done with other forces. If we
want to know the effect of several charges on another charge, we’ll use Coulomb’s law between
the charge we’re interested in and all others to find the forces involved, then we’ll get
For fun, let’s find the gravitational force between two protons and the electrostatic force between
them. The charges are both 1.6 x 10-19 C, the masses are both 1.67 x 10-27 kg, and we’ll put them
10-15 m apart (i.e., next to each other in the same nucleus). We get:
FGrav =
(
)(
)(
)
)(
)
G m1 m 2 6.67 × 10 −11 N m 2 / kg 2 1.67 ×10 −27 kg 1.67 × 10 −27 kg
=
= 1.86 ×10 − 34 N
2
2
−
15
r
10 m
(
while for the electrostatic case, we have:
(
)
)(
k q1 q2 8.99 ×109 N m 2 / C 2 1.6 ×10 −19 C 1.6 ×10 −19 C
FElec =
=
= 230 N
2
r2
10 −15 m
(
)
Look at the difference! The electrostatic force is 1036 times larger! That’s a trillion trillion
trillion times. This is roughly the same size difference as the diameter of an atom compared to
the diameter of the entire universe!
The Electric Field
Sometimes, the concept of a field is a useful one. We’ve already used this idea when talking
about the weight of an object on Earth’s surface. What we’re really doing is finding the force of
gravity between some small mass and the rest of the Earth. Since, in so many cases, we know
that the separation between the centers is essentially just the Earth’s radius, we can do some of
the math in advance and multiply the Earth’s mass by G and divide by the square of Earth’s
radius. After doing that, if we want the weight of an object on Earth’s surface, we just multiply it
by the number we found above (which we call “g”). This g is sometimes called the gravitational
field of the Earth. All we need to do to turn the field into a force is to multiply by the mass we’re
interested in. Since we know force is a vector, and mass is a scalar, the field must be a vector
(pointing into the Earth).
We can use the same idea to find the electric field produced by a charge. Now it will be force
divided by charge rather than mass. We’ll use the letter E for the electric field, and measure it in
Newtons per Coulomb (i.e., if the field strength at a point is 25 N/C and we place a charge of
10-15 C at that point, the force on it should be 25 x 10-15 N). Charge is a scalar, so this field is
also a vector. We’ll define it as pointing from positive to negative charges. In this picture, a
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proton will look like a pincushion with electric field lines streaming out of it in all directions.
Electrons will look the same, but the lines will be pointing inward. We define it this way so that
the force will equal the charge times E. For another electron, the charge will be negative, and
therefore E must be negative also to get a repulsion.
Electric field lines
Isolated positive charge
Isolated negative charge
The restriction on this idea is that we can detect the magnitude & direction of the electric field by
measuring the force on a charge as we move it around in the field, but we need to use a small
charge. If we’re trying to find the field of a 10-17 C charge, we won’t be able to do that very
easily if we’re using a 10 C charge to do it! The small charge’s field will become completely
overwhelmed and dominated by the field of the large charge. For this reason, we will usually
talk about the motion of a test charge in the electric field. This is a charge which is tiny on the
scale of the charge producing the field we want to measure.
We do this so that we can get a more general result than Coulomb’s law alone would give us.
We avoid many long calculations for the gravitational force by defining g, and we’ll avoid a
similar amount of work by defining E. Just as g is independent of the mass we’re measuring, E
is independent of the magnitude or sign of the test charge we use.
For a point charge, then, the E field looks like this:
→
E=
kq
r2
Other shapes have other fields. If we arrange two metal plates so that they are facing each other,
we can use this setup to separate and store charge. We’ll have positive charges on one plate, and
negative on the other. This is called a parallel-plate capacitor. Capacitance is basically the
ability to store charge. When we pile these opposite charges on the plates, an electric field will
develop (pointing from the positively-charged plate to the negatively charged one). The field
lines will be perpendicular to the plates, so that our configuration looks like this:
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+
+
+
+
+
+
+
-
Using a method we’ll examine shortly, we can find that the electric field inside the capacitor is
given by
E=
σ
ε 0 A ε0
q
=
where q is the total charge on each plate and A is the area of each plate. Interestingly, the
strength of this field doesn’t depend on the position between the plates – it’s no larger close to
one plate than in the center. In the real world, the field will bulge out a little near the edges of
the plate.
What are the rules for drawing field lines? First, we know they go from positive to negative
charges. Because of that, they can’t stop in empty space. Also, since they represent the path a
positive test charge q would follow, they can’t cross (which way would it go at the crossing?).
For larger charges, we should draw more lines and put them closer together. If we put a positive
charge near a negative charge, they will be connected by many field lines, and many others will
go out and terminate on (or start on) far away charges. Your book has a picture of this
configuration, known as a dipole (because you have two opposite poles near each other). If the
two charges in a dipole are the same magnitude and opposite sign, the total charge in that region
is zero. This doesn’t mean that the field around the charges is zero everywhere, though. The
reason is that a point will be equally far away from both charges only on a plane between the two
charges, and even there, the field won’t cancel because of the different signs. As you get far
away from the dipole, the field gets small (faster than the field of a single point charge). Where
the field of a single charge falls off as the inverse square of your distance to it, a dipole field
drops off as the inverse cube of the distance. The reason is that, as you get farther away from the
two charges, the difference between your distance to the positive charge and your distance to the
negative charge approaches zero. If the distances were exactly the same, the charges would
cancel and we’d have no field!
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Notice that at every point, the electric field is tangent to the field lines.
Conductors and Shielding
If there is a net charge on a conductor (too many or too few electrons), the charges will all go to
the surface in an effort to get as far away from each other as possible, since like charges repel and
conductors allow charges to move freely. What does that mean for the electric field inside the
conductor? You can answer this by thinking about it either of two ways: first, if there was an
electric field, the charges in the conductor would move in response to it. The electrons would
move in a direction opposite to the electric field, and each one moving that way would act to
reduce the electric field, until the field was zero (this happens in a very tiny fraction of a second
in a conductor – essentially instantly). The other way you can see the field must be zero is to
imagine the charges on a conducting sphere. They will be spread evenly along the surface, with
no particular point singled out. How would the electric field know which way to point?
If there is a cavity inside the conductor, there will be no electric field inside it. This provides a
way to shield a region from external electric fields. This kind of sheltered environment is called
a Faraday cage. If we put a conductor into an external electric field, it will alter the field
slightly. Wherever the field touches the conductor, it will be perpendicular to its surface. It has
to be – a component along the surface would push electrons around in the conductor until they
built up and cancelled that part of the field. Everything we’ve talked about depends on the free
movement of charges – if we’re in an insulator, we can’t assume any of this is true.
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Gauss’ Law
One of the most significant parts of electricity and magnetism is Gauss’ law. This law explains
the electric field produced by a collection of charges, which is very important. We need to know
what the field is if we want to know how things move in it. Basically, Gauss’ law says that we
can draw any surface we want in space – it can be a sphere, cylinder, cube, cow, whatever – and
it can enclose charges, or not. If we take the charge enclosed and divide it by ε0, we will get the
electric flux through the surface. The flux is just the electric field multiplied by the surface area
and the cosine of the angle between the electric field and the normal to the surface. For the cow
shape, it’s an ugly problem. If we stick to things that are a little more symmetric, the math is
much easier. In a sphere, for example, E will always be perpendicular to the surface, so the
cosine term will go away. If we put an imaginary sphere of radius r and area A (which must be 4
π r2 ) around a charge q, we get the formula below:
flux = Φ = E A =
q
ε0
We know everything in this equation except E, which we’re trying to find. Plug in the formula
for the area of a sphere, and solve for E:
E=
q
4π ε0 r 2
We know F = q E, so if we bring another charge (call it q’ ) close to this one, the force will be
F = q' E =
q q'
4π ε 0 r 2
which we recognize. If there’s no charge in the sphere we drew, the flux through it is zero.
Notice that this isn’t the same as saying the electric field is zero – it just says that the amount of
electric field coming in matches the amount leaving. If we draw the sphere around a positive
charge, the E-field is all outgoing, or if the charge is negative, it’s all incoming. For this reason,
charges are sometimes thought of as sources (positive) or sinks (negative) of the electric field.
For the charge below, if the surface on which we apply Gauss’ law is the red dashed sphere, we
can see all sorts of arrows leaving it – electric field is flowing out of it, and there is no balancing
electric field flowing in. If our surface is instead the blue dotted sphere, as much electric field is
flowing out of the right side as is entering the left side. The net flux is zero, because the
enclosed charge is zero. (If the charge below were negative, everything would look exactly the
same except for the direction of the arrows, which would be reversed).
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Potential Energy
In the same way that a mass in a gravitational field has potential energy equal to mgh, a charge in
an electric field has a potential energy. The mass would “like” to move in the direction of the
gravitational field, and the charge would like to move either in the direction of the field (positive
charge) or opposite to it (negative charge). Both gravity and the electric force are conservative
(remember that that means a decrease in potential energy is matched by an increase in kinetic
energy), which is why we can define a potential.
As before, work is force multiplied by distance. In the case of gravity, that gave us mgh (near the
Earth’s surface, at least, where we can say that the gravitational field is a constant g). If the
electric field is constant, we’ll just get the expression below
W AB = q E ( a − b )
where a and b are the initial and final locations of the charge q. As we’d expect, the work done
depends on the size of the charge, just as the work done by gravity depends on the mass which is
moving from place to place. We’d like to eliminate the dependence on the test charge. We do
that by defining an electric potential which is related to, but not the same as the electric
potential energy. All we have to do is divide both sides of the equation above by the charge q.
This will prove to be so useful that we should give this new quantity (work divided by charge) a
name of its own. We’ll call it V for voltage and the units will be joules per coulomb, or volts.
Part of the value of this will be that we’ll automatically know the change in a charged particle’s
energy when it goes between two points of different voltage if we know its charge. For this
reason, we’ll sometimes refer to voltage as the potential difference.
An electron moving from the negative pole of a battery (AA, AAA, C, or D) is accelerated
through a potential difference of 1.5 volts. That means that each electron will experience an
increase in energy of
W = qV = (1.6 ×10 −19 C ) (1.5V ) = 2.4 ×10 −19 J
This is a tiny amount of energy, of course. A more convenient unit of energy when we’re
actually interested in very tiny things like electrons is the electron volt. This is easier because an
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electron moving through the 1.5 V potential difference would have a change in energy of 1.5
electron volts. A little bit of math will show that 1 electron volt = 1.6 x 10-19 J. If we want, we
can find the increase in an electron’s velocity when it goes through a given potential by assuming
that the potential energy becomes kinetic energy.
While electrons generally do the moving when there is a potential difference applied to
something, it was originally believed that positive charges were moving to cause the electric
current. Electrons are said to have a negative charge, though, so the flow of electrons is from a
negative battery terminal to a positive battery terminal. When we look at currents, we’ll
generally refer to them as moving through a circuit from the positive terminal to the negative
terminal, even though we know that’s not exactly what’s happening.
Potential Difference created by a Point Charge
We shouldn’t be very surprised that a point charge produces its own electric potential as well as
responding to the potential in its area. We generally ignored it last semester, but a mass moving
in the gravitational field of a planet will also create its own gravitational field. For both masses
and charges, something that can sense the field (and move in response to it) will also create its
own field.
Before we define this potential difference, we should recall something from our discussions of
gravitational potential energy. We claimed that it was just mgh. We know, though, that work is
force times distance. What we’re saying when we use mgh is that the force (mg) is independent
of distance, so we can find work by simple multiplication. If h is very small when compared to
the Earth’s radius, this approximation will be fine. We don’t use the same formula when looking
at the energy necessary to send a rocket to the Moon because the Earth-Moon distance is very
much larger than the Earth’s radius. The acceleration due to gravity is not g (9.8 m/s2) when we
get far from Earth.
Why do we have to worry about this with the electric force? While we have to find a very large
collection of matter before gravity becomes an important force, a significant electric charge can
be contained in a very small region. When the charge is in a small region, moving a small
distance from that charge will significantly change the field. In other words, it is relatively
common to work with electric fields which can’t be considered constant over the distances that
interest us, and we can no longer just multiply a single value for the field by the distance a charge
is moved in that field to find potential energy.
As you might guess, doing this requires calculus. We need to find the area under a graph of force
vs. distance, and that will give us work (or energy). Another similarity between the electric force
and gravity is that motion around the charge or mass is “free”. Just as the potential energy
doesn’t change when we move a mass along the ground at a constant height, we won’t see the
electric potential energy change if we move around a charge without changing our distance to it.
We used a trick to find the potential energy of a spring (our last force which varies with distance)
by making a graph of force vs. distance, noticing that it was a triangular shape, and using the
formula for the area of a triangle to get energy. We don’t have quite the same situation here, but
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we can quote the result from calculus. As in the case of gravity, the potential energy change of a
charge depends on two points (start and finish). We get, after doing the math,
k q1 q 2 k q1 q2
W AB =
−
rA
rB
Notice that we have two charges here – one creating the field and one moving in it from A to B.
It’s also symmetric, so the potential energy of q1 moving in the field of q2 is the same as the
potential energy of q2 moving in the field of q1. If we replace k with G and charges with masses,
it will look just like the formula for the gravitational potential energy of a satellite.
To go from potential energy to potential, we just drop out the test charge (I’ll call q2 the test
charge and drop it, even though we’ve already said there’s no difference in the math):
V B −V A =
k q1 k q1
−
rB
rA
By convention, we place one of the points (B) at infinity, which makes the term involving rB drop
out. We then refer to the potential at a point r from a test charge as
V=
kq
r
Notice that the potential is a scalar. We don’t have to worry about components here. This can
be a real simplification when we want to see how a test charge moves under the influence of lots
of other charges. We don’t have to find the vector components of the electric field from each
charge and then combine them – we can just add the potential difference produced by each one at
the location of our test charge to find the total potential difference. The potential produced by a
charge will have the same sign as the charge. If we have a positive charge and a negative charge
near each other, there will be a place on the line connecting them where the potential is zero.
Keep in mind that that doesn’t mean a charge placed there would stay at rest – the direction of
movement depends on the way the potential changes with position and the sign of the test charge.
Equipotential Surfaces and Electric Fields
Just as we saw with the gravitational field, a charge will only do work (or have work done on it)
when it gets closer to or further away from another charge. When we move a mass around on the
ground (assuming g is constant and the Earth is a perfect sphere), we don’t do any work against
gravity. If we move one charge around another charge, keeping the distance between the two
constant (tracing out the surface of a sphere), we won’t do any work. From the formula above
for potential, every point on the sphere’s surface will have the same value of V. For that reason,
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we call it an equipotential surface. We can move a charge anywhere on an equipotential
surface without doing any work. You’ve seen this idea before – on a topographic map, contour
lines connect points of equal height. You could move a mass around this line without doing any
work. Also, you see the same thing on weather maps – contour lines connect points of equal
pressure (isobars). When lines like this are closely spaced, that means there is a more dramatic
change in what they measure than in a region where the lines are far from one another. When the
lines on the weather map are closely spaced near you, there will be high winds (driven by a
dramatic change in pressure over a small distance). When there are closely spaced lines on a
topographic map, you have a very steep incline to go up (or down).
What do we get when lines of equipotential are closely spaced? A large electric field! We can
define the electric field as the something called the gradient of the electric potential. The
gradient is a measure of the direction of the most rapid change of something. If you’re on a hill
and set a bowling ball down, it will roll in the direction where the height change is most rapid. If
we let go of a charge at some point, it will move in the direction where the electric field is largest
(so the potential is changing most rapidly). Since negative charges move in the opposite
direction that the electric field points, we need to define the electric field like this:
E =−
∆V
∆s
where ∆V is the change in potential between two points and ∆s is the distance between the two
points.
In the drawings above, we have the electric field lines on the left and the equipotential surfaces
on the right. The electric field is much stronger between the two charges where the equipotential
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surfaces are close together than far away from the charges. Notice that the electric field is
perpendicular to the equipotential surfaces everywhere. For this reason, also notice that we never
see equipotential surfaces cross (since each represents a different potential, what potential would
we have at the crossing? And what way would a charge move at that point?).
In a conductor, all points are at the same potential (so it’s kind of an equipotential volume rather
than surface). The reason is the same thing that makes it a conductor – the charges inside it are
free to move in response to an electric field. If there were a field of some kind in the conductor,
the charges would move in response to it. Once enough electrons moved in the direction
opposite to the field, they would create a field of their own which would cancel out the original
field. It’s the same idea as assuming all points on the surface of the water in a bucket are at the
same height – if there were a small hill or valley in the water, it would very quickly be erased by
the movement of the rest of the water.
Capacitors and Dielectrics
We’ve seen the parallel plate capacitor before. In this arrangement, each plate will have a charge
of the same magnitude but opposite sign. The formula we use to determine the amount of charge
on each plate is just
Q=CV
where Q is the charge per plate, V is the voltage of the battery, and C is called the capacitance.
This factor contains information about the geometry and construction of the capacitor, and the
units of capacitance are called farads (1 F = 1 Coulomb/Volt). Farads are huge units of
larger, the amount of charge a battery of a given voltage can separate increases.
An electric field will be developed between the plates of the parallel-plate capacitor, and it is
(approximately) uniform. Its magnitude will be E = V/d where d is the distance between the
plates. At the edges of the plates, the field will get a little strange, but we’ll generally ignore that.
The electric field will point from the positive plate to the negative one, as always.
The purpose of a capacitor is to store charge so that it can be released (usually much faster than it
was stored) later. For a large value of capacitance, a small potential difference between the
plates will allow us to separate a large amount of charge. For this reason, we would like to find a
way to increase the capacitance of a capacitor if possible.
The formula for C for parallel-plate capacitors is just
C=
κ ε0 A
d
We’ll talk about what κ is later.
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It turns out that we can do this by filling the space between the two conductors (we’ll use a
parallel-plate model again) with a non-conducting material called a dielectric (what would
happen if we used a conductor?). When a dielectric is placed between the plates, the electric
field will tend to line up the molecules in the dielectric in such a way as to oppose the capacitor’s
field. Imagine a molecule (or atom) as a collection of positive and negative charges (protons &
neutrons). When placed in an electric field, the electrons will want to move towards the source
of the field, since the source is a positive charge, and the protons will want to move in the
direction of the field, since field lines end on negative charges. Therefore, the capacitor’s field
tends to stretch the atom very slightly. When the positive and negative charges are separated,
even if only by a tiny amount, we get a dipole which has its own field. This field will act in a
direction opposite to the capacitor’s field, and the overall effect is to produce a smaller total field
between the plates. See the figure below for the effect.
-
+
+
+
+
-
+ + -
+
+
+
+
Dipoles (ovals) form in dielectric (green) in
response to field of plates. E-field of
dipoles (red) opposes field of plates and
makes total internal field (blue) smaller
Total internal
The field between the plates (E) is reduced compared to its value when the space between the
plates is empty (E0) by an amount κ, where κ is called the dielectric constant and depends on the
composition of the dielectric itself. Our formula for κ is therefore:
Strong E-field
between plates
κ=
E0
E
Reducing the size of the electric field by placing this material inside the capacitor acts to increase
the capacitance of the arrangement. In other words, the same battery can now separate more
charge because of the presence of the dielectric.
Since a battery had to do work to move charges from one plate of a capacitor to the other, energy
is stored when charge is separated. The work done to move these charges is not constant, but
depends on the potential difference between the plates. Each electron that’s moved changes the
potential difference between the plates slightly, and more work must be done to move the next
one. We can find the total work in terms of the total charge and the average potential difference,
which is just half of the final potential difference (since we started at 0V). The total work done,
and therefore the energy stored, is just W = Q * (½ V), but we know that Q = CV, so we can write
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1
U = CV 2
2
where U is the energy stored in the capacitor. We could find the energy density in the region
between the plates by replacing C with the formula for it in terms of its geometry and dielectric
constant, and replacing V with E d. Making those changes to the formula above will still give us
total energy, but if we then divide by the volume inside the plates (A d ) we’ll get an energy
divided by a volume or an energy density:
1  κ ε0 A 
2

( E d)
U
2
d 
u=
= 
Volume
1
2
= κ ε0 E
2
Electromotive Force and Circuits
If we want to talk about charges moving, rather than sitting still, they need a reason to move.
That reason is called the electromotive force or emf. Emf is not really a force, though – it’s just
a potential difference between two places (voltage) which, as we know, will cause charges to
move in response to it. The symbol for emf is a script capital “E” or E. If we use a battery to
provide the emf, it works by means of a chemical reaction which causes electrons to collect at
one end and be depleted from the other end. The potential difference between the ends, or emf,
is a measure of how badly the electrons “want” to get back together with the ions. If you connect
the ends of the battery with a wire so that the electrons have a way to do it, many of them will
move along the wire to rejoin ions (don’t do this – it will drain your battery, and for reasons we’ll
see later, it will also make the wire dangerously hot). For our purposes, a battery is just a source
of emf.
When we place the battery in a circuit, it will push electrons from its negative end through the
circuit (doing some work, in general) and back to the positive end. This flow of charges is called
the current and is measured in coulombs per second, or amperes. For example, your car battery
may be able to deliver several hundred amperes of current to your starter. That means that if you
picked a point somewhere along the battery cable and could count charges, you’d see hundreds of
coulombs of charge go by every second. The formula connecting charge, current, and time is
I=
∆q
∆t
A battery produces direct current (dc), which means the current is always in the same direction
(from the positive pole to the negative pole – this is called the conventional current, and the
idea dates from a time when people thought positive charges were doing the moving. Today we
know that it’s generally the negatively-charged electron which does the traveling, and the flow of
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electrons is therefore opposite to the direction of the current). The wall outlets provide
alternating current (ac) in which the electrons are pushed back and forth, changing their
direction many times per second.
Ohm’s Law
The flow of charge through a circuit is very much like the flow of water through pipes. The
battery is like a pump which pushes water through one end of the pipes and pulls it out the other
end. Just as water doesn’t flow through the pipe without friction, electric charges don’t move
through ordinary conductors without some loss of energy. This loss is called resistance for the
obvious reason that the movement of charges is being resisted by the material they’re moving
through. For many materials, the resistance of a given size of material is a constant and we get
the famous equation shown below:
V=IR
This is known as Ohm’s Law after the person who discovered it. This says that as the potential
difference across a piece of material increases, the current through that material increases in
direct proportion. If we connect the two ends of a 1.5 V battery to a black box (we don’t know
anything about what’s inside – it just has two wires sticking out of it) and get a current of 5
amperes, we can then connect it to a 9 V battery and know that we’ll get a current of 30 amperes
if the box obeys Ohm’s law. Not everything obeys Ohm’s law!
The units of resistance are volts per ampere (V/A) or ohms. The common symbol for the ohm is
the Greek capital omega, or Ω. When we’re selecting wire for a circuit, we generally want it to
have the lowest resistance possible (and we’ll typically assume that the resistance of the wire is
zero unless we have a particular reason to worry about these tiny resistances). What would our
equation above predict for the current produced if we connect the poles of our 9 V battery with a
wire of zero resistance? The equation would predict an infinite (or more accurately, undefined)
current. What stops this from happening is the fact that the battery has internal resistance of its
own, so we don’t have to worry about the math failing. In reality, the internal resistance is
usually pretty small, so you can get currents that are very large by “shorting out” the battery.
Doing this with a car battery can result in a free trip to the hospital.
Circuit Diagrams
It’s easier to draw a picture of an electrical circuit than it is to list every element by name, so we
should learn how to do that. Right now, our circuits will be pretty simple. See the table below
for pictures
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Element
Symbol
Battery
+
-
Capacitor
Resistor
Straight Wire
(zero resistance)
If the circuit doesn’t connect one end of the battery to the other end, it will be open, meaning
there is no path for charge to take to get from one end of the battery to the other.
Resistance and Resistivity
The resistance of a given object depends on a few factors: first, the composition of the object.
Wires are made of copper, not glass. This measurement of the inherent ability of a material to
conduct electricity is called resistivity and is usually abbreviated by the letter ρ. For two objects
made of the same material but which have different sizes or shapes, resistance will also be
different. Resistance is proportional to the length of the wire (or object) and inversely
proportional to its area. In other words, a fatter pipe lowers the resistance, and a longer pipe
increases it. The formula for resistance can be written as
R=
ρL
A
where L is the length of the current path through the object and A is the area of the wire. This
tells us that the units of resistivity must be ohms per meter. From the table in your book, you can
see that some metals like copper and silver have resistivities lower than 2 x 10-8 Ω*m. At the
insulating end of the spectrum, resistivities are in the neighborhood of 1014 Ω*m or higher. This
is why wire is typically made of copper (good conductor) covered with plastic (very bad
conductor). If the cord for your lamp was not insulated, the two wires would eventually bump
into each other and provide a way for the charges to return to the wall without lighting your lamp
(short circuit). If you happened to step on both wires, the current would go through your foot!
The dependence of resistance on area can be seen when we look at the sizes of wire used in
various situations. The wires inside your radio or computer are typically carrying very small
currents and therefore can be very small themselves. The main power cable into your house is
probably about the size of your wrist since it’s carrying more current. This is why you
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sometimes hear that extension cords can be dangerous. If you’re using a thin cord (high gauge),
you’re putting a resistance in between the power and the device. You can usually feel the cord
getting warm. Longer cords should also be fatter to counteract this problem. For a
demonstration of this, go to a hardware store or home center and ask them for a 50-foot extension
cord you can use with your stove or clothes dryer!
What causes resistance? It’s collisions between the moving electrons and the rest of the wire
(including impurities in the wire). Imagine rolling bowling balls down hills covered with tree
stumps. The balls will roll a short distance and then collide with a stump, losing speed and
possibly going back uphill briefly. They’ll get downhill eventually, but it will depend on the
spacing of the tree stumps. Current in a wire is also sometimes compared to water in a hose. An
interesting similarity between the two is that, in both cases, the speed of the particle (water or an
electron) is very much less than the speed at which the signal travels. As soon as you flick a light
switch, the electricity travels with a speed not far from that of light. The individual electrons
have speeds (because of all the collisions slowing them down) which are typically a few
centimeters per second or so. The speed difference in a hose is also noticeable – if the hose is
filled with water already, it will start spraying almost as soon as the faucet is turned on. An
individual water molecule, though, takes a while to get from the faucet to the end of the hose.
The resistance of most substances is temperature-dependent. At higher temperatures, the
resistance increases. For some materials, they can be cooled to a point where all resistance to the
flow of current disappears. This is called superconductivity and was discovered in the early
1900’s, but the highest temperatures at which it could be observed were in the neighborhood of
20 K. In the mid 1980’s, new kinds of superconductors were discovered which remained
superconducting to around 175 K or so. This is still very cold, but it’s much easier to cool
something to 175 K than 20 K! A material that is superconducting at room temperature would be
a major breakthrough and would save an unbelievable amount of energy every year which is now
lost to heat.
Electric Power
We can find the power consumed in a circuit ( = the power delivered by a battery) by taking
advantage of what we know about energy. We know that moving a charge q through a potential
difference V requires an energy equal to qV. We also know that energy per time is power. We
can combine these to get
P=
∆q
V =IV
∆t
The unit of power is still what it was before – 1 watt = 1 J/sec or, in this case, 1 Volt-Ampere. If
the device we’re looking at obeys Ohm’s law, we could also write:
V2
P= I R =
R
2
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Using this, what is the resistance of one of the heating elements in an electric water heater if it
consumes 2500 W at 220 V? It’s about 20 Ω. That means the current must be around 11.2 A.
Alternating Current
Instead of a battery, which produces direct current, most of our power comes from generators
(we’ll see how they work later) which produce alternating current (ac). AC means the potential
difference between two points (the plug holes in the wall) oscillates back and forth from some
maximum value through zero on to the negative of the maximum, and back. In the US, a
complete cycle of voltage takes 1/60 of a second, so we say that that our power system is 60 Hz
AC (it’s 50 Hz in most of Europe). The voltage at any point in time can be written as
V = V0 Sin ( 2 π f t )
where f is the frequency in Hertz (multiplying by 2 π converts it to radians per second) and t is
the time we’re interested in. If the circuit is made up of resistors only, the current will change
sign at the same time the voltage does (known as being in phase). Therefore, the changing
current can be written as
I=
V V0
= Sin ( 2 π f t )
R R
Power is still I V, so we’ll get a sin from both terms and that gives
P = I 0 V0 Sin 2 ( 2 π f t )
A plot of Sin2 t vs. t is shown below
As you can see, the power will range from a maximum of I0 V0 to zero. The average of Sin2
over a complete cycle is just ½. The average power is then ½ I0 V0 . We can preserve our old
form of the equation for power in terms of current and voltage if we define a new current and
new voltage. We can split the factor of ½ up between current and voltage and divide each one by
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the square root of 2. The new quantities are called the root mean square or rms current and
voltage and are connected to the maximum current and voltage I0 and V0 by
I rms =
I0
2
and Vrms =
V0
2
so the average power is just Pavg = Irms Vrms and we can use all of our earlier expressions for DC
currents, voltages, and resistances by just replacing I’s and V’s with their rms equivalents.
The 120 Volt figure commonly used to describe ordinary electrical outlets in the US refers to the
rms voltage. That tells us that, at its peak, the voltage should be larger by a factor of the square
root of two, giving about 170 V.
Resistors in Series
If we have more than one resistor in a circuit, one way we can combine them is to join them at
one end. If we do this, all of the current must pass through each resistor. This obviously acts to
increase the resistance of the circuit. There will be a drop in voltage from one side of a resistor
to the other. The size of the drop is determined by the current flowing through the resistor and its
resistance.
In the diagram above, it is clear that the current goes through each resistor. In this situation, the
resistance of the two resistors is equal to the sums of their individual resistances. In other words,
RTotal = R1 + R2 + R3 + 
for multiple resistors in series. For example, if the battery produces a potential difference of 18
V and the resistances of the two resistors are 2 Ω and 10 Ω, what will happen in the circuit? The
total resistance, by our formula above, must be 2 Ω + 10 Ω = 12 Ω. That means the current must
be V/R or 18 V / 12 Ω, or 1.5 A. (Think about what would happen between the resistors if the
current were different in each!). For a current of 1.5 A, the voltage drop across the 2 Ω resistor
will be 2 Ω times 1.5 A or 3 V. When we get to the next resistor, the voltage drop across it will
be 10 Ω times 1.5 A or 15 V. Total voltage drop = 15 V + 3 V = 18 V = battery voltage, so
we’re OK. If you’ve seen old Christmas lights, they were built so that one blown light bulb (the
filament of which is just a resistor) took out the entire string. That’s because they were wired in
series. If you break one part of the circuit, it’s all going to stop.
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Resistors in Parallel
If the resistors in a circuit are arranged so that there are multiple separate paths for the current to
take, they are said to be in parallel.
In the drawing above, a given electron will pass through either one resistor or the other, but not
both. This provides a second path for the current to take and therefore reduces the total
resistance to its flow. If you imagine people leaving a crowded building, we can think of them as
the charges and resistors as the doors. If we have two doors in parallel (meaning you can go
through either one to get outside), more people will be able to leave per second than with only
one of the doors open. Putting doors in series would correspond to having to use both doors to
get out, and that will obviously let fewer people leave than if they only had to go through one
door or the other.
The voltage drop across each resistor above is the same as the voltage of the battery (they’re both
connected to the poles of the battery individually, if we consider the wires to have zero
resistance). For different resistances, these identical voltage drops mean different currents pass
through each resistor. If we return to our 18 V battery and resistors of 2 Ω and 10 Ω, we’ll find
that the current through the 2 Ω resistor is 18 V / 2 Ω or 9 A. The current through the 10 Ω
resistor is 18 V / 10 Ω or 1.8 A, giving a total current in the circuit of 10.8 A. If we want to
replace this pair with a single resistor which gives the same current flow, it would have to be 18
V / 10.8 A or 1.67 Ω. We can find that directly by learning how to combine resistors in parallel.
The equivalent resistance of a number of resistors in parallel is
1
RTotal
=
1
1
1
+
+
+
R1 R2 R3
Let’s check this formula on the result we just found. We should find that 1/RTotal is just the sum
of 1 / 2 Ω and 1 / 10 Ω, or 6/10 Ω. RTotal must then be 1/ (6/10 Ω) or 10/6 Ω = 1.67 Ω. (This,
by the way, is also the way to determine the mpg a van has to get before it will use less gas on a
trip than 2 (or more) cars – remember that for spring break).
To summarize, resistors in series all carry the same current, but have voltage drops which depend
on their individual resistances. Resistors in parallel are all at the same voltage, but carry a
current which depends on their individual resistances. The resistance of resistors in series is
always greater than the largest individual resistance. The resistance of resistors in parallel is
always smaller than the smallest individual resistance.
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Now we see why short circuits are dangerous. If the current can go through a wire laid across the
battery terminals, the resistance in the wire is very low – close to zero. The circuit has the same
resistance it always had. The current now has two choices. It can take a low resistance path, or a
high resistance path. It will take the low resistance path and lots of current will flow. We know
that the power lost to heat in this wire (because it has some resistance, even if it is small) will be
V2/R or I2R, which will be very large when R is small.
How are the outlets in your house wired? They must be in parallel. If they were in series, all of
your lights and appliances would go off when a light bulb blew. This also explains why plugging
too many appliances into a circuit will cause your circuit breaker to trip. Each device you add in
parallel lowers the total resistance of the circuit and therefore increases the current. When the
current reaches a certain value, the circuit breaker will trip and the power will be cut off. This
used to be done by fuses (and is still done that way in a car). The idea behind the fuse is to
isolate a small piece of wire made of a material which melts at a low temperature. If that little
isolated piece of wire burns through, the circuit is open. It’s much better for that little piece of
safely-contained wire to melt than for a random part of the wire inside your wall to do it!
Many of the resistive circuits we’ll eventually work with can be reduced to collections of
resistors in series and parallel. Whenever you’re trying to decide whether two resistors are in
series or parallel, just look at the current path: if the current is required to go through both,
they’re in series. If it’s required to pick a path, they’re in parallel.
Internal Resistance
As mentioned before, a battery (or a generator, for that matter), has some internal resistance of its
own. This is usually very small, but because it will effectively be placed in series with the rest of
the circuit, there will be a voltage drop across it. The true voltage across the terminals is called
the terminal voltage, and will be lower than it would be without the internal resistance. As the
resistance of the rest of the circuit decreases, the importance of the internal resistance increases.
Higher current through that internal resistance means a higher voltage drop across it.
Kirchoff’s Rules
For complicated circuits, which may have more than one battery or generator and multiple
current paths, we need to use a couple of guiding principles to figure out what’s really going on.
These are called Kirchoff’s rules, and the reasoning behind them will be easy to see. First, the
junction rule. This says that, for a given junction (place where 2 or more wires connect), the
incoming currents must exactly match the outgoing currents. If this were not true, charge would
pile up somewhere in the circuit, causing an emf which would oppose the driving emf. This is
really just a statement about the conservation of charge, and we can return to our analogy of
people in a building. You can be sure that the number of people entering a revolving door is
equal to the number of people leaving it. Mathematically, the currents entering a junction will be
positive and the currents leaving it will be negative. The sum of these positives and negatives is
required to be zero (Rule 1).
The loop rule just tells us that energy is conserved. If we follow a charge all the way around a
circuit, it will see as many potential drops (across resistors, etc.) as potential jumps (across
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batteries or generators). We’ll say that the end of a resistor closest to the positive terminal of a
battery or generator is itself positive, and the other end is negative. If there is more than one
source of emf, we just guess at the direction of current flow and mark our resistors so that it
flows from + to -. When a charge (here we’re assuming positive charges to make life simpler)
moves from the positive side of a resistor to the negative side, there is a potential drop equal to
IR. When it moves through the battery from the negative side to the positive side (after passing
through the rest of the circuit), it will experience a jump in potential equal to the battery’s
voltage. The sum of potential drops and potential jumps through the circuit will be zero.
We have a little freedom with this method, since we can pick the direction of the current if there
are multiple sources of emf and we don’t really know which way it is going in the loop. We
combine our two rules, applying the junction rule at each junction and the loop rule around every
loop. We’ll get multiple equations with multiple unknowns, and we then have to solve them. If,
at any point, we get a negative value for one of our currents, it just means we picked the wrong
direction for it.
Current and Voltage Measurements
How can we check ourselves and see if the numbers we calculate for things like currents and
voltage drops are accurate? As we’ll see later, a current both responds to and creates its own
magnetic field. By arranging a coil of wire between magnets, we can watch it move as a current
passes through it. If we put a spring on it to resist its movement and a needle on the coil to track
the movement, we’ve made a device called a galvanometer, which is the main part of a current
measuring instrument called an ammeter. The other important part is a shunt resistor, which
gives the current an alternate path through the device and thereby diverts large currents (most of
them, at least) around the sensitive galvanometer. If the galvanometer needle deflects enough to
measure small currents, we can see that very large currents would cause the needle to go around
many times (or, more likely, would burn out the thin wires inside the galvanometer). As long as
we know the relative resistances of the galvanometer and the shunt resistor, we can figure out the
total current.
The ammeter has to be inside the circuit to measure the current since it can only measure what
flows through it. That means it will be in series with the part of the circuit it’s trying to measure.
That also means that its presence will increase the total resistance and therefore lower the
current. The process of measurement alters the true value we’re trying to measure! This is not
really unfamiliar to us, though. If we want to know the temperature of a small cup of coffee and
we stick a large, room-temperature thermometer in it, we know from last semester’s study of heat
that the coffee will cool a little as it warms the thermometer up. We minimize this effect in the
specific heat lab by using quantities of water which are large compared to the mass of the
thermometer. We can minimize the effect of our ammeter by making its internal resistance as
small as possible.
If we connect the galvanometer in parallel with part of the circuit, we will be providing an
alternate path for the current. If the galvanometer has a tiny resistance, we’ll siphon off most of
the current and burn out our ammeter. To get around this, we put a large resistor in series with
the galvanometer to make the current path less attractive to the charges in the main circuit. The
current that does pass through the galvanometer (deflecting the needle) can be multiplied by the
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total resistance of the galvanometer + large series resistor to get the voltage between two points.
Now we’ve made a voltmeter. Because the addition of a voltmeter gives a new current path, it
will alter the behavior of the circuit unless the resistance in the new path is very large. Therefore,
we’d like a voltmeter to have a very high resistance, just like we want our ammeter to have very
low resistance.
Remember that ammeters are wired in series with the circuit, and voltmeters are wired in parallel
with it. The ammeter can be thought of as replacing a tiny segment of the wire in the circuit.
The voltmeter is connected between two different points to measure the potential difference. If
we do this backwards, we’ll have a problem. A voltmeter in series with a circuit will make the
total resistance huge and will probably stop the current. An ammeter in parallel with a circuit
will (because of its low resistance) suddenly find itself routing a huge current between two
points. For this reason, most ammeters have fuses which are designed to burn out first before the
galvanometer can be destroyed.
Capacitors in Series and Parallel
Just as we have more than one way to put a resistor in a circuit, we have more than one way to
wire capacitors in a circuit. As before, if the capacitors are all connected to the same two points
(like across a battery), we’ll call that a parallel arrangement. Take a look at the drawing below
for an example:
-
+
These capacitors are connected in parallel, but how are they different from a single capacitor with
area equal to the sum of their areas? There is no difference. For a given voltage V, the charge on
capacitor 1 is Q1 = C1V and the charge on 2 is Q2 = C2V. The total charge is therefore Q1 + Q2 =
C1V + C2V = (C1 + C2)V. The capacitance of the two in parallel is then just the sum of their
individual capacitances: CTotal = C1 + C2. Capacitors in parallel behave kind of like resistors in
series.
If the capacitors are in series, interestingly enough, we’ll see that they combine like resistors in
parallel! Notice that the right plate of the left capacitor and the left plate of the right capacitor in
the drawing below are connected by a wire.
-
+
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The plates must have charges which are equal in magnitude but opposite in sign. That means
that the plates facing them also have charges of the same magnitude and opposite sign. We’ll
call it Q and label them all
+Q -Q +Q -Q
-
+
Since the voltage is split between the capacitors, the sum of the voltages across each should be
the battery’s voltage.
V = V1 + V2 =
Q Q
Q
+
=
C1 C2 CTotal
As you might have expected by now, this means that the capacitance of several capacitors in
series is found by the same math we use to find the resistance of several resistors in parallel:
1
CTotal
=
1
1
+
+
C1 C 2
RC Circuits
We’ve glossed over this before, but when we connect a capacitor to a circuit, the charge that it
separates doesn’t collect instantly. As charges flow from one end to the other through the
battery, they collect on the plates and begin to establish an electric field that opposes the
accumulation of more charges. This shouldn’t be surprising – we know that the negative charges
on one plate would like to be as far from each other as possible. The charge on the capacitor
builds up to its theoretical maximum value of Q0 = C V as time passes according to the formula
below:
(
Q( t ) = Q0 1 − e −t / RC
)
In other words, the rate at which the capacitor charges depends on the resistance and capacitance
of the circuit. How long would it take to fully charge? According to the formula (and assuming
charge doesn’t come in fixed-size units of electrons), an infinite amount of time. As a practical
matter, we’ll see that Q is close enough to Q0 after a few time constants have passed. The time
constant for the circuit is found by just multiplying resistance and capacitance:
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τ = RC
After one time constant (so t = τ ), the value of the charge will be Q0 (1 – e-1) or about 0.63 Q0.
When t = 5 τ, we’ll get Q0 (1 – e-5) or about 0.993 Q0. The capacitor will discharge exponentially
as well. Assuming it starts off charged to Q0, it will discharge according to
Q( t ) = Q0 e
− t / RC
The combination of resistor and capacitor makes a useful device for measuring time
electronically.
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