Electrostatics 2014
Level 1: Basic Charges
The universe is made up of basic particles that combine and separate to form all matter. These basic particles
(as you learned in chemistry) consist of protons, electrons and neutrons. Electrons are small masses with a
negative charge. Protons have a positive charge and a comparatively larger mass. Neutrons have the same
mass as protons, but no charge. For the sake of this course, we will largely be dealing with protons and
electrons. Neutrons are rarely mentioned (until we get to quantum dynamics much later in the year).
Charge on the Particles
We measure charge in Coulombs (C). The smallest charge in the universe (that we will deal with in this class) is
1.6 × 10−19 C. We call this the elementary charge, because the rest of the universe is built upon positive and
negative multiples of this charge. The negative elementary charge is the electron, which a charge of −1.6 ×
10−19 C. The positive elementary charge is the proton with a charge of +1.6 × 10−19 C. We symbolize an
elementary charge with a lower case e. When we talk about charge, an electron, therefore, is –e and a
proton is +e.
Think About It
If an object had 3 electrons and 2 protons, it would have a total charge of –e. If an object had 100 protons and 98
electrons, it would have a charge of +2e. State these charge in terms of Coulombs.
Most of the time, matter does not have any charge. This is because the protons and electrons perfectly cancel
out. In some situations, however, an atom might gain or lose electrons. If an object lost some electrons, it has
more protons than electrons, and therefore has a positive charge. If an electron has gained electrons, it has an
overall negative charge. Notice we never discuss gaining or losing protons.
Key Idea
Objects only charge through the movement of electrons, not the movement of protons.
A typical atom has the same number of protons and electrons, so it is effectively neutral. Sometimes an atom will
have an unbalanced charge, however, because it lost or gained an electron in a chemical process. This is called
ionization. An ion is just a term for any atom that has a charge- this could mean it has a slightly positive charge
or a slightly negative charge.
The most important thing you need to remember throughout this entire section is the Law of Conservation of
Charge.
Key Idea
The Law of Conservation of Charge states charges cannot be created or destroyed. They can be
exchanged, passed around, added and subtracted to objects, but overall, you can’t destroy charge any more
than you can destroy matter.
Practice Problem: Finding the Number of Charges
Some materials, called conductors, allow charges to move around rather freely. Insulators are the opposite. If
you stick a bunch of electrons on an insulator, it will keep them pretty much where you left them. Think of
insulators as prisons, where you lock the charge in a particular cell. Think of conductors as pools of water,
where the particles can swim away from each other (or toward each other) like fish.
Key Idea
Opposite charges are attracted to one another.
This means protons and electrons are drawn together.
Objects with opposite charges are attracted to one
another. How attracted they are depends on the how
much charge is involved and how far apart they are.
Key Idea
Like charges are repelled by one another.
Protons are repelled by one another. Electrons will always try to move away from each other when given a
chance, and the same thing happens with protons. Objects with similar charges are repelled by one another.
Again, how repulsed by one another they are depends on how much charge is involved and how far apart they
are.
If you put a bunch of protons on a conducting sphere. In a fraction of a second they will move like this:
Inside a Conductor
Because the charges what to get as far away from each other as possible, they will always spread out
along the outside of a conductor. This means that you will never have a charge inside on a conductor. It
you put one there, it would immediately transfer to the outside.
Pop Quiz: Would You Survive?
you are in the middle of an Illinois field (very very flat) when a terrible lightning storm blows up. Where is the
safest place to take shelter- under a very tall tree, in a nearby, quickly flowing stream, or a nearby chicken
coop made of chicken wire?
Answer
Let’s explore what the best answer would be.
Because the charges what to get as far away from each other as possible, they will always spread out along the
outside of a conductor. This means that you will never have a charge inside on a conductor. It you put one there,
it would immediately transfer to the outside. Therefore, the chicken coop is your best option.
Key Idea
There is never a net electric charge inside a conductor.
A Faraday Cage is a fancy name for a conductor that is hollow inside.
If you place an object inside, it would essentially be the safe from any
charge you would zap the outside with. The picture on the left is
completely real. Many science museums have Faraday Cages like this
one set up for shows. You should, of course, never ever try this on
your own.
A simple faraday cage can be created out of nearly any conductor.
There is a growing market for fashionable Faraday Cages, as they will
essentially make your phone unable to receive a signal.
Supplemental: Fashionable Faraday Cages. Here is one example of people marketing
this idea.
http://www.technologyreview.com/view/421768/silence-smart-phones-at-thanksgivingdinner-with-a-foldable-faraday-cage/
Conduction
There are a variety of ways to charge objects. The most common is conduction. Conduction (just like in
thermodynamics) involves charging by touching. This is the type you loved as a kid, when you rubbed a
balloon all over your head and pulled it away, watching your hair stick to the balloon. In this situation, you
ripped electrons off your hair and transferred them to your balloon. Afterward, your (positively charged) hair
was attracted to the (negatively charged) balloon.
Induction
Because particles in conductors move away from like charges (and move toward opposite charges) we can
mess around with them a little. Imagine you have a metal sphere with positive and negative equally balanced
on its surface. This sphere currently has a neutral charge (see the picture below on the left). Now, imagine that
you brought a wand with a positive charge nearby (without touching it). What would happen?
The protons can’t move- they are stuck in place. But the electrons they are paired with can flow around in a
conductor. If you bring a positively charged rod nearby, the electrons will be drawn to it. They will leave their
protons and rush toward the wand.
This metal sphere now has one side positive and the other negative. The second you move the outside positive
wand away, however, the positives and negatives would mix up again, and the sphere would go back to being
neutral. This charging by separating charges (without touching) is called induction.
What if we wanted that sphere to keep that charge? What if we wanted the object to permanently have more
electrons than protons? To do that, we hook the sphere up to a ground. A ground is pretty much a path (often a
wire) that connects to the ground. The earth itself is a vast sea of positive and negative electrons. You can toss
electrons into it, or draw electrons up out of it.
When the wand is brought near, the
electrons in the sphere rush toward the
wand (like the previous scenario). Now, let’s
imagine we attach a ground to the sphere.
Electrons can leave the earth and flow into
the sphere- which they are drawn to do by
the attraction of the positively charged
wand. Once these electrons are up in the
sphere, we disconnect the ground and move
the wand away. The sphere has a (fairly)
permanent negative charge.
The same can be done to get the sphere to
have a positive charge. In this case, we
approach the sphere with a negative wand.
The ground allows the electrons to run away from the negative charges on
the wand (left).
A quick note: object that have a positive charge on one end and a negative
charge on the other (like the sphere when the wand was brought near) are
polarized.
Important: Before moving on to the video, make sure I have stopped by
your table and shown you the electroscope.
Video Simulation: This video will help you visualize what is
happening during a few complex electroscope demos. Make sure
you watch it until you understand it.
http://www.youtube.com/watch?v=-JsVZwc1dOo
Level 2: Coulomb’s Law and Linear Superposition
We’ve talked about the fact that opposites attract and likes repel. In this section, we’ll learn how to calculate
how to calculate the strength of that attraction or repulsion. What if, for example, we had two charges hanging
out in space, a certain distance apart? How much attraction/repulsion would they feel?
That’s right, folks. We’ve finally made it to an equation.
Coulomb’s Law
|𝑭 =
𝒌𝒒𝟏 𝒒𝟐
|
𝒓𝟐
q Charge, measured in Coulombs
r  Distance between the charges, measured in meters
k  Another sweet constant. This one is 9 x 109 N m2/C2
See the line things on either side of the equation? Those mean absolute value. That means that when we
calculate the force, we will always write the number as a positive. We do this because if we kept the positive or
negative, it would imply a direction, and this equation does not tell us the direction the force is acting. You
will have to figure that out through reasoning- are these two like charges? Then they repel. Are these two
opposite charges? Then they attract. Let’s look at at a simple example.
Example Problem
from The Physics Classroom
Suppose that two point charges, each with a charge of +1.00 Coulomb are separated by a distance of 1.00 meter.
Determine the magnitude of the electrical force of repulsion between them.
Solution
Felect = k • Q1 • Q2 / d2
Felect = (9.0 x 109 N•m2/C2) • (1.00 C) • (1.00 C) / (1.00 m)2
Felect = 9.0 x 109 N
The equation above can only tell you the magnitude (the amount) of the force. It does not tell you the direction of
the force (attraction or repulsion). When you calculate the force, you should always use the absolute value. You
are going to need to keep track of whether it is repelling or attracting yourself using free body diagrams. Later
on in this section, we will look at how to use trig and free body diagrams to keep track of more complex
problems. For now, when you enter the charges into the equation, use their magnitudes (amounts, without the
sign) and then ask yourself whether this would be a force of attraction or repulsion.
Practice: Newton’s 3rd Law
Consider the following situation… +Q has twice the charge of +q. They are both positive. For each of the
particle, draw a free body diagram showing the force acting on them.
Solution
Did you draw your free body diagram with two identical (but opposite) forces acting on each of the charges?
Good. This is accurate because, according to Newton’s 3rd Law, the force exerted by +Q on +q is exactly the
same as the force +q exerts on +Q. This is similar to the way the earth pulls down on you with approximately
600 N. You, in turn, pull up on the earth with a force of 600 N. Why doesn’t the earth rise up to meet your feet
when you jump, then? Just because two objects feel the same force does not mean they react to it the same way.
To you, with your relatively puny mass, 600 N enough to smash you into the ground. To the earth, with its
enormous mass, 600 N is nothing. Similarly, just because two particles have the same force acting on them does
not mean they will necessarily react the same way.
Example Problem
Solution
What Superposition Means
Of course, it couldn’t be that easy. Brace yourself, because this will be the
most math-y thing we do all year. While you will sometimes encounter
problems involving two objects attraction/repulsion from one another, you
will much more often encounter problems like the one on the right.
Forces (like the forces of electrostatic repulsion/attraction) are vectors,
which means we need to take their direction into account when we are
adding them together. Superposition is just a fancy way of saying “adding
vectors”. Hold on to your hats, kids, because this section is going to take a
little trig.
Linear Superposition
Before we dive into the trig, let’s look at a problem where the particles are in a straight line. When you solve
these, draw a free body diagram. Just a refresher, check out the free body diagrams below and make sure you
are clear on where the net force came from.
Now, try a problem involving Coulomb’s Law. Remember to draw a free body diagram.
Example Problem
Solution
Supplemental: Watch Twu Solve Linear Superposition Problems This is not a required video,
but if you are someone who likes to watch people solve problems, give this one a try.
http://www.youtube.com/watch?v=bB4PdHUimZc
Level 3: Superposition in 2D
Before we move on, make sure you find the net force in the situation below.
That wasn’t too bad, huh? Okay, now let’s add in some trig. Remember to draw a free body diagram and break
the vectors down into x and y components. If you have trouble remembering how to add vectors from last year,
call me over and I can help you review. This is going to require a blast from the physics past and quite a bit of
trig.
Trig Review Looking for a quick trig refresher? Just want to take a break and watch a trig-based
rap? Here you go.
http://www.youtube.com/watch?v=t2uPYYLH4Zo
.Review Problem
DO NOT move on until you feel completely comfortable doing this. Remember that a net force includes a
direction.
a) Add the forces together and find the net horizontal force.
b) Add the forces together and find the net vertical force.
c) Find the net force including the direction.
Solution
Try this yourself before checking the solution below.
First, break F1 and F3 down into their components (the parts of them that point in the x and y direction). I’ve
shown you how to do this with F1 to help jog your memory
𝐹1𝑥 = 𝐹1 cos(55)
𝐹1𝑥 = 7.3 cos(55)
𝐹1𝑥 = 4.18
𝐹1𝑦 = 7.3 sin(55) = 5.98
When you’ve done this for all the forces, add (and subtract) the forces to help you find the net force in the x and
y directions. The work below shows this.
The forces in the x and y directions are therefore:
Use Pythagorean to find the net force. You should get 11.3 N. But wait, there’s more. You need to use trig to find
the direction of the net force. Opposite over adjacent, son. If you did this right, you should get 22.7°.
Example Problem
The distance between the 2nC and 3nC charge is 5 cm. Using the diagram above, find:
a) The total horizontal forces acting on the 4nC charge.
b) The total vertical forces acting on the 4 nC charge.
c) Find the net force on the 4nC charge.
Solutions
Example Problem
Princeton Review
Solution
Brace yourself, it is about to get real. Don’t look at the solution to this one until you have tried it on your own.
from Princeton Review
Solution
Level 4: Electric Field Strength
Electric field strength is not a particularly difficult concept, but it is very abstract. I’ve taught this for years and
I’ve only found one way that (sometimes) clicks with students. You will have to bear with me, because it is a
little weird.
Electric Field Strength gives us an indication of how attractive (or repulsive) a charge would be if another
particle was placed nearby.
It does not tell us about the force of attraction between two particles. It tells about one particle’s ability to attract
or repel at a particle position.
Here is the analogy I’ve used in the past which has helped some people.
Imagine you are standing a particular distance away from an attractive person. I’ve put both genders here, so
pick. Imagine someone you think is pretty attractive.
You are standing a set distance from this attractive person. If you were both electrostatic particles, Coulomb’s
Law would tell us how attracted you are to each other. We could calculate the force pulling you two crazy kids
together (or pushing you apart, depending on your charge).
But what if we took you out of the picture entirely? This person is still attractive right? In this analogy, Electric
Field Strength would let us calculate how attractive that person is in general (not just to you).
Person is
attractive in
general, not
just to you
Key Idea
Electric field strength isn’t about the relationship between two particles- it describes the
attractive/repulsive ability of a single particle at a particular distance away from it. It’s about how that
particle influences the space around it.
The Test Particle
How do we calculate exactly how attractive/repulsive a particle is? Use the equation below.
Electric Field Strength
𝐸=
𝑘𝑞
𝑟2
E  Electric field strength, measured in N/C
q  charge of the particle
k same constant as before
r  distance from the charge.
Let’s try a basic electric field strength problem, plug n’ chug…
Practice Problem
What is the electric field strength 2 cm away from a 4 µC charge?
Answer
9 × 107 𝑁/𝐶. Don’t forget to convert that 2cm into meters.
What did we just calculate? We basically calculated the attractive/repulsive ability of the 4 µC charge 2 cm
away. This tells us amount of force this object can exert per coulomb. That’s what the answer 9 × 107 𝑁/𝐶
means. If we put a 1 C charge at this spot (2 cm away), it would feel a repulsive force of 9 × 107 𝑁. If we put a 2
C charge at this point, it would feel a repulsion of 18 × 107 𝑁. If we put a 3 C charge at this spot, it would feel a
repulsion of 27 × 107 𝑁….
In other words, the electric field strength gave us an indication of exactly how attractive/repulsive that 4 µC
charge would be 2 cm away.
What if we put a charge of 6 µC at a position of 2 cm away from the 4 µC charge? Is there a way we could use the
electric field strength to calculate the force of repulsion between the two particles? Sure! Here it is:
𝐹 = 𝐸𝑞
E Electric Field Strength of a particle at a particular position
q  charge of a different particle
F Force of attraction/repulsion between the particles
Example Problem
A +10µ C charge is placed in an electric field with a strength of 20, 000 N/C at the location of the charge. When
released, how much electric force will act on the charge?
Solution
Once you know the electric field strength at a particular position, it is very easy to calculate the force that would
act on particles that are put at that spot.
Adding Electric Fields
Electric fields can be added much the same way that electric force can be added. Both are vectors- which
means you need to take their direction into account when you are adding them. The problem below
demonstrates how this works.
from Holt Physics
Example Problem
Solution
Summary of What We’ve Learned So Far
Calculates a relationship between two
charges
𝑘𝑄𝑞
𝐹= 2
𝑟
To convert
𝐸=
𝐹
𝑞
Describes something about one
charge
𝑘𝑄
𝐸= 2
𝑟
Level 5: Visualizing Electric Fields
Electric field strength is a vector, which means you need to take its direction into account (just like with forces).
When we try visualize electrostatic forces, we often create free body diagrams. When we want to visualize
electric field strength, we draw electric field lines. They look like this:
In this section, we are going to learn how to create and interpret electric field line diagrams. This might seem
complicated, but it actually isn’t.
The Test Charge
Let’s image we have some large, random charge floating around in space. We want to figure to visualize what
the electric field strength would be around the charge. How could we do this out?
In physics, we get an idea of the electric field around it by seeing what happens to a test charge.
A test charge is just a tiny hypothetical charge we will move around nearby to see how it would respond. The
test charge itself won’t actually be in the problem- it is just a tool we use to check for the electric field strength.
Think of it as your electric field meter man, walking around the charge in question, testing out the electric field
strength at various positions.
A test charge is, by definition:


Incredibly small and massless, so it won’t influence the problem.
Always positive. This is just a convention all physicists agreed to long ago, like deciding down was
negative and up is positive.
As you move the test charge around, ask yourself two questions:


Which way is the electric field pushing the test charge?
How hard is it pushing it (really hard, sorta hard, not that hard…)?
Electric field diagrams are qualitative (at least at this level) which means we don’t care about numbers. We
care about the general shape.
Let’s do this one together. On the picture below, draw an arrow showing the direction the electric field would
push this charge. Use an arrow to show how strong the test charge would be.
If you did that right, it should look this this:
Now, repeat that process for all the positions I have marked below. Remember, longer arrows mean greater
electric field strength.
Check with your group members to make sure they have
something similar to you.
You can see there is a general shape to the electric field above.
creates an electric force that pushes the test charge (which is
away. On the right, I show how we would draw the official electric
charge.
The charge
positive)
field for this
Use this same logic to create the electric field diagram for a
charge. Try to draw that below:
negative
If you did it correctly, it would look like this:
Now, all electric field lines are drawn as though they are attracting or repelling a
positive test charge. What would happen if you were to put an electron in the
electric field to the left? What would it do? What if you put an electron in the
electric field to the left- what happens?
Key Idea:
Positive charges move in the direction of the field lines.
Negative charges move against the field lines.
As much as I’m sure you’d like to sit around drawing field lines for individual
particles, I’m sure you know what is coming…. field lines for multiple particles. Before you move to the next
page, try to draw the electric fields for the situations below. Imagine, again, what happens to test charges in
order to figure this out.
If you did these correctly, they look this this:
Interpreting Electric Field Lines
In order to interpret electric field lines, it’s important to know a few things about them. You may have already
noticed some of these key features already. You will never have to memorize the list below, but taking time to
learn them will make your life a lot easier (a lot like memorizing thermo processes made them a lot easier).
Here are some basic rules for drawing/interpreting electric field lines:
1. Electric field lines do not cross.
2. Electric field lines come out of positive charges and go into negatives (think about the test
charge).
3. The denser the lines, the greater the
electric field strength.
from Princeton Review