Download PHYS 202 Notes, Week 1

PHYS 202 Notes, Week 1
Greg Christian
January 19 & 21, 2016
This week we introduce the fundamental quantities of electric charge
and the laws governing the forces and fields generated by electric charges.
Electric Charge
Electric charge is a fundamental property of matter. It’s something that
is intrinsic to everything, similar to mass. Only two types of charge exist: positive (+) and negative (−)1 . Opposite charges (+/+ or −/−)
repel each other, while like charges (+/−) attract. Electric charge is
conserved, that is, it cannot be created or destroyed. However, we can
move charges around to create a net charge on a given object2 . The total
amount of net charge on an object is called its magnitude, while the sign
tells whether it’s positive or negative.
As named by Benjamin Franklin in the
1700s.
1
A classic example is rubbing a glass rod
with silk, giving it a net positive charge,
or a plastic rod with wool giving a net
negative charge.
2
7
Li Atom
Net charge zero
Where does electric charge come from?
Electric charge originates in atoms, from the three types of particles
of which they are composed: protons (positive charge), neutrons (no
charge), and electrons (negative charge). Protons and electrons have
the same magnitude of charge but opposite signs.
As pictured in Figure 1, these particle are arranged to make atoms
in the following way:
• A nucleus made of protons and neutrons at the center with net
positive charge.
• Electrons surrounding the nucleus with net negative charge.
-
-
n+n
n
+ +
n
-
nucleus
(charge +3)
Figure 1:
An atom, made up of
a positively-charged nucleus and
negatively-charged surrounding electrons.
In normal atoms, the number of protons and electrons is the same,
giving the total atom a net charge of zero. In other words, ordinary
atoms are charge neutral. Likewise, so is most ordinary matter.
The electrons in atoms are only weakly bound to the nucleus. As
a result, it is possible to add or remove them, creating ions, which do
have a net charge. There are two types of ions:
1. Positive ions created by removing electrons.
2. Negative ions created by adding electrons.
Drawings of each are shown in Figure 2. In a normal-sized (or macroscopic) object, made up of many atoms,3 large numbers of electrons can
be added or removed. This gives the object a net charge. In the classic
example of rubbing a glass rod with silk or a plastic rod with wool,
this is what’s going on:
electrons
(charge -3)
3
On the order of 1023 .
phys 202 notes, week 1
(a) 7Li negative ion
Net charge -1
-
-
electrons
(charge -4)
(b)
7
Li positive ion
Net charge +1
-
nucleus
-
Figure 2: a) A positively charged ion,
with electrons removed; b) A negatively
charged ion, with electrons added.
nucleus
n + n (charge +3)
n
+ +
n
n + n (charge +3)
n
+ +
n
-
2
electrons
(charge -2)
-
• Rubbing the glass rod with silk removes electrons, giving a net positive
charge.
• Rubbing the plastic rod with wool adds electrons, giving a net negative charge.
Conductors and Insulators
In general, materials can be divided into two types with respect to
whether or not charges can move through them4 :
1. Insulators, which do not allow charges to move.
2. Conductors, which do allow charges to move.
When a conductor connects two objects with different charges, it allows that charge to move between the two objects. In contrast, when
an insulator connects two differently-charged objects, the charges stay
where they are. A good example of this is a wire connected to a battery. The wire is a conductor and allows charge to move freely from
the positive terminal of the battery to the negative one. This moving
charge is what powers things like a lightbulb or a smartphone or a
Tesla car. However, the plastic surrounding the wire is an insulator
that does not allow charge to move through it.
In conductors, the moving charge results from either electrons or
ions moving through the material. Whatever is moving through the
material is sometimes referred to as the charge carrier. Most conductors
are metals, such as copper wire, whose charge carriers are electrons5 .
The free movement of charge through conductors can lead to some
notable effects.
Induction
The free movement of electrons in a metal6 leads to a phenomena
called induction. This results from the electrons moving from one area
of the conductor to another, leading to local excesses of positive and
There is also a third type, the semiconductor, which can behave like an insulator or a conductor depending on the situation
4
The other common types of conductors
are liquids or gasses which allow ions
(which can be either positively or negatively charged) to move. A good example is seawater, whose charge carrier is
the dissolved salt ions
5
6
Or ions in some other conductor.
phys 202 notes, week 1
3
negative charge. Figure 3 shows an example of this. Here’s a breakdown of what happens:
• The negatively charged rod repels the electrons on the surface of the
metal, creating a force towards the right side of the sphere.
• The electrons move towards the right side of the sphere, leaving a
net positive charge on the left side.
• The positive charge attracts electrons back towards the left.
• Things end up in equilibrium with the force to the right from the
rod exactly balancing the force to the left from the positive charges.
Figure 3: Example of induced charge on
a metal sphere.
Conservation and Quantization of Charge
There are two fundamental principles relating to charges:
1. The conservation of charge: charge cannot be created or destroyed.
It can only be moved around to create local excesses of positive or
negative charge.
2. The quantization of charge: charge comes in discrete packets, with
the fundamental unit being the charge on an electron (or proton).
Any charge you see is some combination of these packets. Compare
with currency, which comes in discrete packets of 1¢.
Coulomb’s Law
Coulomb’s law7 describes the force F arising from two charges q1 and
q2 some distance r apart:8
F=k
| q1 q2 |
.
r2
(1)
It says that the magnitude of the force is
Discovered in 1784 by Charles Augustin de Coulomb using a torsion balance.
8
Note the similarity to the law of gravity, F = Gm1 m2 /r2 . The only difference
is that mass is always positive, making
gravity always attractive.
7
a) Proportional to the product of the magnitude of the charges; and
b) Inversely proportional to the square of the distance between them.
The direction of the force is determined by the signs of the charges:
opposite signs attract while like signs repel.
In Eq. (1), the term k is a fundamental constant of nature. Using SI
units, it’s equal to
k = 8.99 × 109 N · m2 /C2 .
This introduces the SI unit of charge, the coulomb, denoted as C. The
fundamental unit of charge is denoted by the symbol e and has a value
of9
e = 1.60217653 × 10−19 C.
As alluded to in the last section, the
charge on an electron is −e and that on
a proton is +e.
9
phys 202 notes, week 1
As you can see, this is a very small quantity in terms of Coulombs.
Even compared to charges encountered in everyday life, the Coulomb
is quite large. Charges on the order of microcoulombs (1 µC = 10−6 C)
or nanocoulombs (1 nC = 10−9 C) are more common.
The constant k is often expressed10 in terms of another fundamental
constant e0 :
1
k=
,
(2)
4πe0
where e0 = 8.854 × 10−12 C2 / N · m2 .
4
This may seem like a silly overcomplication at the moment, but later
on you’ll learn that expressing things in
terms of e0 makes sense when studying
different topics.
10
q2 = +1 C
Superposition
Coulomb’s law obeys the principle of superposition, that is, the total
force on a given charge is the vector sum of all the acting forces from
other charges. Figure 4 illustrates an example of this.
Electric Fields
When charges interact by Coulomb’s law, what is there to tell them
about each other? To answer this question, we can introduce the concept of an electric field. The electric field is a quantity describing the
magnitude and direction of the force that would act on a charge q if
it were at a particular location in space. Again, the electric field is an abstract quantity—a concept that we use to describe how real forces are
transmitted.
To define the electric field, pretend there is a “test charge” q0 at some
location P in space. The electric field is given by the force that would
act on the test charge divided by its magnitude:
~E = ~F/q0 .
F12
q1 = +1 C
F23
q3 = -1 C
Ftot
Figure 4: Illustration of the superposition principle. The total force on q1 is
the vector sum of the forces from q2 (repulsive) and q3 (attractive).
(3)
Both the force and the electric field are vector quantities, meaning that
they have both a magnitude and a direction. In Eq. (3), the direction of
~F and ~E is the same if q0 is positive and opposite if q0 is negative.
The electric field is different at every point in space, just how the
force on q0 would be different if you moved it around. This is illustrated in Figure 5. Because it has magnitude and direction, the electric
field is referred to as a vector field.
One thing that is important to remember is that the actual electric
field has nothing to do with the test charge q0 and does not depend on
it in any way. It is simply a device that we made up to illustrate the
concept. Electric fields are, however, generated by other collections of
charges, called the source of the electric field.
Figure 5: Illustration of the electric field
generated by a point charge. The field
gets weaker as you move away from the
charge, just like the force on a test charge
would get weaker as it was moved away
(c.f. Coulomb’s law, Eq. (1)).
phys 202 notes, week 1
5
Calculating Electric Fields
The magnitude and direction of an electric field can be modeled with
mathematical equations. Before getting into this, let’s introduce two
concepts:
1. The source point, S: the location of the charges generating the field.
2. The field point, P: the location where we want to calculate the field.
In general, electric fields can be generated by multiple sources. In this
case, the total field is the vector sum of the individual fields,11
~Etotal = ~E1 + ~E2 + ~E3 + . . .
(4)
The actual equations describing electric fields depend on the geometry of the source. Next, we’ll introduce the equations for some of the
more common source shapes.
Point Charge
This electric field due to a point charge q is given by
E=k
|q|
.
r2
(5)
This follows naturally from Coulomb’s law, Eq. (1), and the electric
field definition, Eq. (3).
Spherical Charge Distribution
Spherical charge distributions have a unique and useful property:
• The electric field outside a sphere containing charge q is exactly the
same as the field produced by a point charge at the center of the
sphere.
In practice, what this means is that we can treat a spherical charge distribution just like a point charge, using Eq. (5) to calculate the electric
field outside it. One thing to be aware of, however, is that this principle
only applies to points outside the sphere.
Example Problems
Coulomb’s Law
What we mean by vector sum, is that
both the magnitude and the direction of
the individual fields are added together.
Think of stacking a bunch of arrows up
together head-to-tail.
11
phys 202 notes, week 1
Electric Fields
9
Opposite of force (b/c negative charge):: UP
DOWN