Understanding Electric Charges and Forces: A Comprehensive Guide
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Introduction
In this comprehensive guide, we delve into essential concepts in physics, specifically focused on electric charges, their properties, and how they interact with each other. We explore fundamental topics such as atomic structure, Coulomb's Law, electric fields, and the dynamics of forces related to electric charges. This article is tailored for students and enthusiasts looking to solidify their understanding of these concepts.
Atomic Structure
At the core of every matter is the atom, which comprises three fundamental components: protons, neutrons, and electrons. The nucleus of an atom contains protons and neutrons, while electrons orbit around the nucleus. Each of these atomic constituents has a specific electric charge:
 Proton (q = +1.6 × 10^19 coulombs): Positively charged
 Electron (q = 1.6 × 10^19 coulombs): Negatively charged
 Neutron (q = 0): No charge
Electric Charge and Coulomb's Law
Electric charge is a fundamental property that determines how particles interact electromagnetically. According to Coulomb's Law, the electric force between two charges can be calculated using the formula:
[ F = k \frac{q_1 q_2}{r^2} ]
Where:
 ( F ) is the magnitude of the force between the charges,
 ( k = \frac{1}{4 \pi \epsilon_0} ) is Coulomb's constant (approximately 9 × 10^9 N m²/C²),
 ( q_1 ) and ( q_2 ) are the amounts of the charges,
 ( r ) is the distance between the centers of the two charges.
The importance of Coulomb's Law lies in its application to various scenarios involving electric interactions. If the charges are of the same sign, the force is repulsive; if they are opposite, the force is attractive.
Superposition Principle
To analyze systems with multiple charges, we use the Principle of Superposition, which states that the total force acting on a charge is the vector sum of all individual forces exerted by all other charges. Mathematically,
[ F_{net} = F_1 + F_2 + F_3 + ... ]
This principle simplifies the process of calculating net forces in complex systems involving multiple interacting charges.
Electric Field
The concept of an electric field (E) expands our understanding of how charged objects influence the space around them. The electric field created by a charge can be defined as the force experienced by a unit positive charge placed in the field.
The electric field is given by:
[ E = k \frac{q}{r^2} ]
Where ( q ) is the charge creating the field and ( r ) is the distance from the charge. The direction of the electric field is outward from positive charges and inward towards negative charges.
Visualizing Electric Fields with Field Lines
Another effective way to represent electric fields is through field lines. Field lines visually represent the direction and relative strength of an electric field:
 Density of lines represents strength: The closer the lines, the stronger the field.
 Direction of lines shows the field direction: Lines always point away from positive charges and toward negative charges.
The Electric Field of a Dipole
A dipole consists of two equal and opposite charges separated by a distance. The electric field produced by a dipole can be calculated at various points, particularly along the axis and on the perpendicular bisector of the dipole. For large distances, the electric field of a dipole behaves differently.

Along the axis: [ E \approx \frac{kp}{x^3} ] (where ( p ) is the dipole moment and ( x ) is the distance from the center)

Perpendicular bisector: [ E \approx \frac{p}{2\pi \epsilon_0 y^3} ] (where ( y ) is the distance from the center of the dipole)
Forces in an Electric Field
When a charged object is placed within an electric field, it experiences a force:
[ F = qE ] Where ( q ) is the charge and ( E ) is the electric field strength at that location.
Torque on a Dipole in an Electric Field
When a dipole is placed in a uniform electric field, it experiences a torque that tends to align it with the field:
[ \tau = pE \sin(\theta) ] Where ( \theta ) is the angle between the dipole moment ( p ) and the electric field ( E ). This torque results in a rotational motion that aligns the dipole with the field.
Conclusion
Understanding electric charges and their interactions lays the foundation for many areas in physics, especially in electrostatics and electromagnetism. Through concepts like Coulomb's Law, electric fields, and the behavior of dipoles, students can grasp how particles interact within the atomic structure and the larger universe. This knowledge not only reinforces classical physics principles but also prepares learners for advanced topics in physics.
Prof: Okay, normally I would ask in a small class if there is something you didn't follow from last time. I'm afraid to do that now, because it's a big class and I
don't know how many things you follow or didn't follow. What I will do first is write down a very quick summary of the main points from last time.
So you should ask yourself, "Did I follow all those things?" And if your answer is yes, then you are fine.
Because I talked about many, many things, but you don't need all of that. So what I'm going to write down right now is the absolute
essentials of last lecture, okay? That's going to be needed for what I do next. First point was: everything is made of atoms.
some things called neutrons, and outside are some things called electrons. That's all the atomic structure we need.
Then we say certain entities have a property called electric charge. The symbol for electric charge is q, and you can put a
subscript to say who you are talking about. So you can say q for the neutron is 0. A q for the electron is 1.6 times 10 to the 19,
and it's measured in coulombs. The q for the proton is reallylet me put it this way, q for the proton is a positive number,
so these minus signs cancel. Now, the importance of the coulomb is that if anything has some coulombs on it, it will interact with anything
else that has some coulombs on it. So that if you have two entities and this one has a charge of q_1 coulombs,
that one has a charge of q_2 coulombs, and the distance between them is r, then the force is q_1q
because it takes too long, but you all know what the answer is. Namely, if you want the force on 2,
due to 1, will be repulsive if q_1 and q_2 are of the same sign and point to the direction joining them.
There is another force law which is r^(2), not so famous, called Hooke's Law, but you're absolutely right.
That's the difference between being Newton and being Hooke. Hooke is known for the r^(2)^( )law. Newton is known for the 1 over r^(2)^( )law.
So this is a very important thing. If you see anything wrong you should stop me, because when I go home, they're going to play the video
for me to watch, and I'm going to see that r^(2). There is nothing I can do until some voice from the back says,
"Hey, send it downstairs", and I appreciate that a lot, okay? It's good, so never hesitate to do that, plus sign,
minus sign, symbols; anything that goes wrong. It also tells me that you're following me. So for all those reasons you should not hesitate to correct
anything, and you should not think that you don't follow it because it's your fault. Probably it is.
You need only one other ingredient. That's the superposition principle. You need that ingredient because we're not going to be
talking only about two charges. We're going to be talking about many charges. And the question is: what will they do when they're
all present, and it's a great blessing that we have the principle of superposition that says that if you've got, say, three charges,
say 1,2 and 3, you want to know the force on 3 due to 2 and 1. You can find the force that 1 would exert.
You can find the force that 2 would exert. We have two vectors, and you can add the two vectors to get the net force.
In other words, the interaction between pairs of charges is insensitive to the presence of any other charges. They go about their business exactly the same way.
That is a principle that is not deduced by logic. You cannot say, "Of course, it had to be that way."
That's not true. It doesn't have to be that way. In fact it is not that way if you make supreme,
I mean, extremely exquisite measurements, but they occur really at the really quantum level. For classical electromagnetic theory it is actually an
experimental fact that you can superpose. All right, if you combine those things you can calculate anything and everything that we will deal with for sometime,
but that's summary ofthis is all I said last time, okay? Now, I didn't do one thing, which is to emphasize to you
that the force of gravitation divided by the electric force is some number like 10 to the 40. I put this twiddle here meaning not exactly.
There are factors of 1 and 2 or 10 missing, but 10 to the 40 is roughly the magnitude of this ratio, and it was done by comparing the force between electron and a
proton. It didn't matter what the distance was because everything goes like 1 over r^(2), so when they take the ratio
that cancels out, but this was really the ratio of things like mass of the electron, mass of the proton, divided by 1 over
4Πε _0; q of the electron and q of the proton. If you put all these numbers in you've got that.
I want to mention one thing. It may be interesting for you to think about. Look at this nucleus.
Nucleus has a lot of protons in it, and according to Coulomb's Law they all repel each other. And the neutrons, of course, don't do anything
because they have no electric charge. So you should ask yourself, "Why are all these protons staying together inside the nucleus if I don't see any
attraction between them?" I see why the electrons are hanging around, because they're attracted to the nucleus,
because nucleus is positively charged, electrons are negatively charged, and Mr. Coulomb tells you they'll be attracted.
What are the protons doing together in that tiny space? That space is like 10 to the 13 centimeters. So have you ever thought of that, or does anybody know why
Prof: There is a strong nuclear force. Okay, that's the answer. In other words, but that answer will lead to
more questions, but I'll first state the answer. There's a force even stronger than the electric force.
You see, this gives the impression electric force is very strong. It is much stronger than gravity, but there is a force
even stronger than the electric force that is experienced by protons and by neutrons. In other words there is another charge which is not electric
charge, which the protons are endowed with and the neutrons are endowed with. And the force due to that charge, if you like,
it's not called charge but it's a similar thing, is much, much stronger. It had to be, because you have to beat the
electrical repulsion. So then you can ask yourself, "Well, in that case, how did we ever find the
electrical force," because here's another force even stronger than the electrical force, maybe a thousand times stronger.
Student: It's irrelevant over larger distances. Prof: Yes, so let me repeat what he said.
It has to do with what's called the range of the force. In other words, look at two different functions, okay?
One function looks like 1 over 137 times 1 over r^(2). Other function looks like 10 times e to the r over r_0 divided by r^(2);
r_0 is some length, and the length is roughly 10 to the 13 centimeters, or 10 to 15 meters.
Which force are you more impressed with is the question, okay? If r is much bigger than r_0,
say 10 times bigger, you've got e to the 10 on the top. e to the 10 is a small number.
e to the 3 is like 1 over 20. e to the 10 is 1 over 20 cubed. So then what will happen is, this force, even though there's
a number 10 in front of it, will be negligible compared to this one. Whereas if r is much smaller than
r_0, say, r is .1 of r_0, so you can forget this number, e to the minus point is roughly 1,
then this 10 will dominate the 1 over 100. So what happens is, if you're sitting inside the nucleus, the nuclear force is
numerically strong, not only because of the number in front of it, but also because this exponential factor has not kicked in.
Therefore, the protons feel an attraction for each other due to the nuclear force that is stronger than the repulsion due to the coulomb force.
And the neutrons attract the protons just as much as the protons attract the protons with respect to the nuclear force. So neutrons are actually good, because when you throw an extra
neutron into an atom you don't add to the coulomb repulsion, but you add to the overall attraction that they all feel for each other due to the nuclear force.
So neutrons are like the glue. As you add more and more protons, you will find that you've got to add more and more neutrons to compensate the
coulomb attraction. But a time will come when the nucleus becomes so big, that even if you add enough neutrons,
the repulsion between protons from one end of the nucleus to the other is now becoming comparable to the attraction due to the nuclear force, because nucleus has become so
big this factor is no longer negligible. See, over long distances a coulomb force will always triumph, because no matter what the
prefactors are, the exponential factor in front of the nuclear force will always weaken it. That's why you cannot have nuclei beyond some size.
If you make them any bigger, the nuclear electrical repulsion between the distant parts of the protons, due to the distant protons in the nucleus,
cannot be compensated by the shortrange attraction. So it's the range of the interaction that is a significant factor here.
So all the strong forces are strong, but at short distances the coulomb force is not that strong, but it falls like 1 over r^(2).
Now gravity, on the other hand, is exactly 1 over r^(2)^( )with a number that's much, much, much smaller than this,
but then we saw the other day why gravity managed to survive, because this is q_1q _2 and the q's can be added
algebraically and cancel each other. Whereas if you've got m_1m _2_ over r^(2)^( )there is no way
to cancel the m. This is how the different forces manage to survive for the different reasons, okay?
Nuclear wins in the nuclear zone, but dies very quickly outside the size of a nucleus. Electrical forces falls more slowly at like 1 over
r^(2). They dominate atomic physics, but once you formed an atom you've got pretty much an electrical neutral thing,
and once you've got many, many atoms making up our planet, then all that remains is the gravitational attraction between a planet and another planet.
Okay, so these are examples of different forces and why they were found at various times, because they all dominate under different circumstances.
All right, so today I'm going to start with my new stuff. So all you need to know, really, if you want to do your problem sets and your homework, is Coulomb's Law,
you know, how to stick the numbers in the Coulomb's Law. And the only thing I didn't mention is this 1 over 4Πε _0 is 9 times 10
to the 9^(th). So today we're going to do something which is a part of a great abstraction and it goes as follows: So I'm gong to take two
So let's say this guy is at vector r_1, this one is at vector r_2. Now, I will really take are of all the arrows.
And you've got to say force on what, so I'm going to say force on 2, due to 1. Now, you've got to realize it's a convention.
So I use a convention: This is the force on this guy and the force due to this guy, with the second later. Now, let's write it out in detail.
Then here I want to write (r_1  r_2)^(2). That's the 1 over r^(2), but then I have to make it a
vector. So for the vector part, once you have the magnitude of the vector, you should multiply by vector
of length unit 1 going in the direction of this difference vector. So you can use any symbol you like.
One is to say e_12. e's always going to be unit vector going from 1 to 2, or if you're inclined you can also write
They're all unit vectors. Now, do you follow that? I mean, are you having trouble with unit vectors?
Anytime I have a vector pointing from here to there, I want to give a magnitude and direction. The magnitude in this case is 1 over the distance squared,
but you have to append to it a vector of unit length in that direction. That's what makes it into a vector.
For example, suppose I want to describe that vector r, and it is 7 meters long? I cannot write r = 7, because that doesn't tell you
which way it's pointing. So invent the vector called i, which is a unit vector in the x direction and I multiply it by that.
long, okay? So you need to add a vector of unit length to the magnitude, multiply it to get the actual vector.
So that's that formula. Then if you have many charges, I'm not going to do that now, say one more here, it'll exert a force on
I'm just taking two guys. Now, I'm going to formally write this as equal to the electric field at r_2 times
q_2, and this is called a field. You see, if you look at this thing, all I've done is rewrite the expression as something that
involves a charge of q�_2 and everything else that involves the q_1 and the distance from q_1 and
q_2. So it looks like there's no real content to giving this object a name, but it's a very profound
notion, so I've got to tell you the story that goes with it. In Coulomb's Law you say that q_1 and q_2 exert a force on each other,
Now I'm going to say q_1 produces an electric field at the location of q_2 given by this vector, and when I multiply by
Whereas q_1 and q_2 exist only at these two places, the electric field can be defined everywhere.
It doesn't require a second charge because you see, this number you can compute for any value of r_2.
So the picture we have is that q_1 produces an electric field all over space, and q_2 experiences that field and gets repelled as a result,
and the force it feels is the field at that point times q_2. In general, we say there's an electric field at this point in
space. If you put a charge q there it should experience a force q times the electric field.
So to understand this, the charges are only in two places in our example, but the field due to q_1 is everywhere.
will turn into a force if you multiply it by a charge you put at the location of the field. So if you've got one charge here, I claim there's a field
here, there's field there; there's a field everywhere due to this guy. How do I know, because if you put a test
charge it begins to feel a force. So one way to say it, is that the field is the force on a unit charge you put at that location,
unit charge because if q_2 is equal to 1 then numerically E is exactly equal to the force. If I go to you and say, "Find out the electric
What do you think you have to do to measure the field? Yes? Student: Add up all the forces on that field from the
Her answer was, "If I want the field here, I should find out where all the charges are," right?
"Find all the forces they will exert on this guy, on the unit charge here," right? But I'm telling you, that's correct.
That's the theoretical way to calculate the field at that point, but suppose you're an experimentalist and you don't want to know what produced it.
Student: Place a test charge. Prof: Okay, and then? Student: And then see what happens to it.
Prof: By `see what happens', you've got to be moreso his answer was: put a test charge, and see what happens.
Now, you've got to be more precise. I think you know what you meant, but I want you to finish that sentence.
Student: Okay. Well, you can measure the force by measuring the acceleration on the charge.
But you did give the right answer. The answer was, by `see what happens', you put the charge there, and you see what acceleration
it undergoes. Then that acceleration is thethat times the mass is the force it's experiencing.
That should be q that you place there times E. If q was 1 that force itself is equal to E. If q was 10 you've got to divide the force by 10 to get
the field there. So, the field is like the sound of one hand clapping. People say one hand clapping is Zen concept, but the field is
like that, because you don't need two charges to have a field. You just need one charge.
So here is how we understand that. You put a charge q. If you go here something has happened there,
Something really is different at this location because the charge q is present. What is different?
What is different is that when this guy was not here and I put a charge, it just sat there. Whereas when this guy is here and I put a charge,
it experiences a force. So if you put a 1 coulomb here it experiences a force which is 1 times the field at that point, therefore, this charge has
distorted the space around it, in fact, everywhere. Now, if you've got many, many, many charges, then they will all try to produce a force on a unit charge
and you should, like you said, add up the vectors, which are the forces due to all the other charges on that one location,
on a test charge on that location. So the field is the sum of the field due to all the charges at that point.
Now, Coulomb's Law doesn't work when charges are moving. Why is that? Have you any idea why you cannot use it?
Student: r is changing? Prof: r is changing so we'll keep changing it as the charge moves, but it's only an approximation
Student: There's a magnetic field? Prof: True, but even the electric fieldhis answer was magnetic field,
but even the electric field is not properly given. Yes? Student: Is the electric field affected by
something like the Doppler Effect? Prof: Not the Doppler, something else. If Coulomb's Law were exact, okay?
Here is what I can do. You take a coulomb and you sit at the other end of the galaxy. I have a coulomb here.
You know the force my coulomb exerts on yours because it's pushing against you. You hold it.
Now suddenly I move my coulomb away from you by a tiny amount. What will you feel? You will feel the force is reduced.
allow that, because I have managed to communicate with you, arbitrarily far, instantaneously.
The minute I move this charge, you know about it, because your charge moved away. For example, if your charge was connected to
a spring, and it had been extended to some amount because of this charge, the minute I move it your spring will move.
That instantaneous communication is forbidden by the special theory, so it does not happen. So the correct way to do it, it'll maybe come later in the
course, so probably not even at the end of this course, but the proper way to do that is to in the end realize that the electric field at this point
is not only due to what the charges are doing now, but what they were doing in the past. Because if some charge at the other end of the galaxy did
something it takes some time, namely traveling at the speed of light, to carry that information from there to here.
So it's the delayed response to all the motion in the charges that you've got to add to find the field here. That's what makes the computation of the electric
field much more complicated. But if you promise me charges never move then the location now is the location last year, the location a million years
ago, then you can use Coulomb's Law. Coulomb's Law is good for electrostatics, but in real life charges are moving, so you cannot really use
the formula. Now, in our room, if you put a charge here and another charge there, if you move this,
that guy will move pretty much instantaneously. That's because the time it takes a light signal to go from here to there is so small, you may treat it as
instantaneous. So Coulomb's Law is used in electrical circuits and so on. You don't worry about the time of transit because it's too
small, but over longer distances, where the time it takes for light to travel becomes nonnegligible, you cannot use Coulomb's Law.
It's not wrong. It is not appropriate when charges are moving. However, it will always be true that if you go to any one point
the force on any charge q you put there is q times electric field at that point, okay? So the electric field notion survives because it doesn't
violate relativity. It says if the field here is so much q will experience the force q times E, but the complication
is what is the field here? Well, it's due to everybody else, and it's not only due to everybody else right now, but everybody else from the
dawn of time because things have been moving and shaking and sending signals to us. We collect all that and see what lands here at this instant.
That decides the field here. So that's the computation of the field, but the response to the field is very easy.
You put a test charge; q times E is the answer. So in modern physics, in theories that are compatible
with the special theory of relativity we break the force into two parts. Charges don't immediately interact with other charges.
Charges produce a field, and the field may even propagate outwards at the speed of light if you make motions, but another charge at the location of that particular
point will respond to the field at that point. So it's not responding to the charge right now. It's responding to the field this charge produced at its
The first part is, find the field due to this charge configuration, that charge configuration, maybe due to various currents, and the second part is,
given the field, find the response of charges to the field. So you understand charges play a double role.
They are the producers of the field. They are also the ones who respond to the field. If you don't have a charge you cannot produce field.
If you don't have a charge you cannot experience it, you cannot play that game. To gain membership into electrostatic interactions
you've got to have charge. So neutrons cannot do that, but they can do other things. Like I said, they take part in nuclear
force, in fact, just as well as protons do. All right, so let's go back now to the simplest problem in the world: the electric field due to one charge.
The formula is very simple. Let's put that charge at the origin. And the electric field due to 1 charge is q over
The field at this point is that magnitude, and I'm going to write e_r_ meaning a vector of unit length in the radial direction.
Once again, I will tell you if you want, you can write that as the position vector divided by the length of the position vector.
They're equivalent ways. If I write it the second way, you've got to be a little careful.
It'll look like qr over 4Πε _0r^(3). Don't get fooled into thinking the field is falling at 1 over
See, if you write it as 1 over r^(2) put unit vector. If you write it as 1 over r^(3 )put the position vector.
They're all saying the same thing. So here you have this formula. If you're a person who likes to work with formulas this is all
you need. You manipulate the stuff on paper, and you add different fields, but people like to visualize this.
So how do I visualize this? That's the real question. So here's a very popular method for visualizing this formula.
You know, you've got, for examplesuppose someone asks you, what's the height above ground level of a certain part of the United States?
Okay, you've got some mountains. You've got some valleys. Well, somebody can give you a function that gives you the
height at any point in the United States, but it's more interesting to have some kind of a contour map that looks like this, right?
And all these contours are different heights. If you've gone hiking, you can see those maps. They tell you pictorially what a certain function is trying to
It's very easy to write down the electric field at one point, namely you take that point, you draw an arrow there; E.
That E is the electric field at that point. So we try to do our best by saying here is my test charge. I'm going to pick a few points, four points,
maybe eight, and I'm going to tell you what the field is at those points. At this point it looks like that.
At this point it looks like that. Here it looks like this. That's already telling you something.
You've got to be very careful on the interpretation. This arrow is not telling you what's happening throughout the length of the arrow.
It's telling you what's happening at the tip. You understand the arrow is in your mind. It's a vector.
It's not really sticking out in space. It's a property at a condition at that point, but we've got to draw it, so we draw it that way.
It doesn't tell you the state of affairs over its length, but only at its tip, at the starting point. Then you can say, "Okay, what happens when I
go further out?" When I go further out, say over here, it's going to be stillif I put a test charge here it's
still going to be repelled radially, but a lot less. So I do that. So I draw arrows at other representative points and make
So you can do this, okay? But now where all do you want to draw these arrows? It's up to you.
You pick a few points. You go to another radius, you draw more arrows. Someone had this clever idea of doing the following.
You can probably guess. Their idea was when I join all these arrows like that, now if you go to a point like this, what have I gained and
Student: The more information is that now you know the direction that it will continue on forever if no other forces act on it, and what you've lost is the
magnitude... Prof: Okay, let me repeat that. That's what you guys should have been thinking in your head.
When I joined these linesby the way, I do want you to anticipate what I'm going to say, because if I'm struck by lightning,
another electromagnetic phenomenon, can you even complete my... Student (Chorus): Sentence!
Prof: Sentence, right. Okay, now you should be able to go a little beyond if I'm doing a derivation.
You've got to be following me, right? That's very important. It's got to be active.
And I sat through a lecture yesterday for an hour. I know it's a very long time. This is what, an hour and fifteen minutes?
The only way you can survive this is if you somehow make it an active event. You've got to do something that keeps you awake during the
That'll make sure also that you're on top of it, that'll make sure that you catch mistakes. Okay.
All right, so here are these lines. As she said quite correctly, previously I knew the field direction only at the chosen points,
but now I know it throughout this line, but I've lost information on the magnitude of the field, because the arrowsthere are no lengths of anything.
These arrows don't have any length. In fact, you can keep drawing more lines if you like. They go like that in all directions.
It basically tells you, hey, the charge is pushing everything out radially no matter where you are. That's the thrust of this picture.
But, due to the miraculous property of the coulomb force, namely that it falls like 1 over r^(2), there is information even on the strength of the electric
field, and that information is contained in the density of electric field lines. And I'll tell you precisely what I mean.
By density of lines, I mean the number of lines crossing a surface perpendicular to the lines, divided by the area of that surface.
Because let's make a convention that we will draw for every coulomb a certain number of lines, 32 lines per coulomb, 32 lines are going out.
I draw a sphere of some radius, 32 lines cross that sphere. I draw a bigger sphere, 32 lines cross this sphere also, but they're less dense,
because the number of lines per area will be some number of lines per charge divided by the area of the sphere, which is 4Πr^(2).
Do you follow that? If you take a sphere, first of all, every portion of the sphere the area that you have is
perpendicular to the lines. So the area intercepts the lines perpendicularly. That's the agreement here.
And you see how many are crossing per unit area. That is going like 1 over r^(2). So these lines naturally diverge and spread out in space
so that the density falls precisely as 1 over r^(2). That has to do also with the fact you're living in three
dimensions. Only in three dimensions where area goes like r^(2)^( )does this spreading of the density of lines coincide with
the decline of the force. So these lines tell you more than simply the direction. They convey to you visually where the field is strong.
Wherever the lines are dense, the field is strong. Wherever the lines are spread apart, the field is weak, and it's a very precise statement.
The only thing not precise is, how many lines do you want to draw per coulomb. That's really up to you, but you've got to be
consistent. Once you give 32 lines per coulomb, then if you've got a charge of 1 coulomb you should draw 32.
If you've got two coulombs, you should draw 64 lines. As long as you do that, the number of lines crossing per unit area will be proportional to the field.
But I'm going to make a certain choice that will make the number of lines per unit area exactly equal to the field, and here is the choice.
It's a choice that makes life simple. Let us agree that 1 coulomb gets 1 over ε_0 lines.
Don't quote the 40 million. It's whatever this thing is. I don't know what it is.
It's a definite number. Then, what's the nice thing? If you've got q coulombs you will have q over
ε_0 lines, and if you take a sphere of radius r you'll get 1 over 4Πr^(2)^( )as the line density,
namely lines per unit area, but that is exactly equal to the strength of the electric field. If you picked a different number like 2 over
ε_0 you will always be measuring 2 times the electric field. The density will still convey the electric field,
namely it'll be proportional to it, but let's make life easy by making it equal to it. This is just a convenience.
Now, we are really set. If you draw pictures this way, you can go as far as you like from this charge.
Simply take a unit area with you. Take a piece of wood 1 meter by 1 meter, put it there, see how many lines cross; that is equal to the electric
field at that point. Okay, so this is the way one likes to visualize field lines. So I'm going to give you some examples.
For a single charge you just draw it that way. For two charges, let's take two charges, a minus charge and a plus charge.
By the way, I'm not going to draw 1 over ε_0 lines per coulomb because it's going to be too many lines, okay?
I'm just going to draw a few so you get the picture. So I'm going to draw four lines right near the charge. You can forget about all other charges in drawing the lines.
Prof: Why does it depend on the one charge? In principle it depends on every charge. Somebody had an answer back there?
Yes? Student: Since it falls as 1 over r^(2)^( )when you're so much closer to one than the other then...
Prof: Right, because the field is 1 over r^(2)^( )and the 1 over r for this guy is going to infinity.
1 over r for this guy is maybe 1 over 1 meter. It's finite. So when you come arbitrarily close to a charge,
it is going to dominate. Well, if it's the only thing in the universe, we know the lines will look like this.
At least they'll start out this way, but soon, of course, it won't go out this way forever, because you will realize there's another charge.
Likewise, it's easy to draw the lines this way. Remember, the lines are coming in because if you put a test charge, it'll be sucked into this.
Test charge is always assumed to be unit positive charge, so the lines will be coming into a negative charge, and leaving, going away from a positive
charge. Now we've just go to do what the agencies forgot to do, which is to connect the dots.
You do this. Now, at some point you'll have to think a little harder, because suppose I go here?
How do I know I should draw the lines this way? If I take this guy here, it will repel it. This guy will attract it, and I add the two,
and get a line in that direction, so you really have to do a lot of work. If you really want this picture to be exact,
you have to compute the vector everywhere, but if you want a sketch, you're allowed to guess, and things look like this.
So this is called a dipole, and this is the field of a dipole. So here's another example.
Both guys are plus. Now what do the lines look like? Again, they will start out this way near the charges,
but now when you come to the midpoint here there should be no electric field right in the midpoint because it's getting pushed equally from both sides.
So the lines, if you think about how they will add, they will do something like this. Okay, look, I'm not going to do a good job for a variety of
reasons, but you can look at your textbook or any other book to see what the lines will look like when you've two plus charges.
If you go a mile away from these two plus charges, what do you think the lines would look like? Yep?
Student: Just like a point charge that's twice as strong. Prof: Right, so that's the intuition you
should keep in your mind. If you go very far from a charge distribution where you cannot look into the details, all you will see is some little
dot that has the entire charge in it, and the lines will be coming out radially. So only when you zoom in you realize, hey,
it's not a single charge 2q, it is two guys of strength q. Finally, let's take a case where this is charge 2q.
This is charge q. Let's say this is charge q. Then some lines will go like this and some lines will run off
to infinity. Here if you go very, very far away from the two charges you'll again see radially outgoing lines,
So this is the example of a dipole. If you've got more charges it gets more complicated, so people don't usually draw them.
There's one example which is pretty interesting. If you've got one plate and another plate, this contains all positive charges,
this contains all negative charges, then the field here will look like this, will go from the positive to the negative plate,
because if you put a test charge between them, it's getting repelled by the positive plate and attracted by the negative plate, so the lines will go from one
to the other. Near the edges they may do something more complicated, but in the bulk they will look like this.
Okay, so now I'm going to do one calculation, which is: what is the actual electric field due to a dipole?
In other words, not just the picture here, where is that? That picture on the left is a dipole.
I'm going to do it quantitatively. So here's my goal: I want to take a minus charge here, a plus charge here.
So today I'm not going to do the field everywhere, because later on I'll show you a more effective way to calculate it, but I'm going to calculate it
at a couple of interesting easy places. In other words, we all know the lines look like this, okay?
But I want to go to some location and find the magnitude and direction of the field. But today I will only find it at two places,
one along the axis at a point x, and one on the perpendicular bisector at a point (x = 0, y), just going to do those two.
So let's see what's the field here, the field at this point? The field at that point, you agree, is going to be entirely in the x direction because this is pushing it,
_0 times that distance squared, which happens to be (x  a)^(2). That's the repulsion due to the charge that's nearer to you.
Then there's the attraction due to the minus q, but it's a little further away, so it looks like a minus sign but it is (x a)^(2).
If a is equal to 0 you get 0. If a is equal to 0 the two guys are sitting exactly on top of each other.
This whole thing fails to be 0 because this a is not 0, and you can understand why. The minute a is not 0 you're closer to one of the two
charges, so that they cannot really cancel each other. So you've got to manipulate this expression, so I will do that now.
a) is (x^(2 ) a^(2))^( )and everything is under squares. In the numerator you've got (x a)^(2) 
Student: Why is it not x  a squared times x a squared? Prof: It is. You are saying why is it not (x a)^(2) times
I don't want you to give up. There is nothing better than shooting me down, but this happened to be correct.
Prof: No, no, no, no. I don't want to rush through this. Anybody have the same problem with this?
Look, also I am doing it fast because this is 958th time I'm doing this calculation, so if you're seeing it for the first time, I've got to slow down.
Then this is really an x a times another x a, and an x  a and another x  a, but without these guys,
I know it's x^(2 ) a^(2 )because I've got two of everything. I squared everything.
So this answer's actually an exact formula of the electric field along the axis of the dipole. But normally what one is interested in is,
when x is much bigger than a. When it's much bigger than a, downstairs you've got x squared.
This could be 1 kilometer squared, a squared was maybe 1 millimeter squared. So in the first approximation, it's not an exact formula
anymore, from now on it is approximate, in the limit x is much bigger than a. You can see it's going to be i times q over
I took from the 4a. I borrowed a 2a to write this here and I canceled the 2 with the 4 here to get that.
So let me writecan everybody see this thing from wherever you are; the last formula here? So I'm going to write it as i times p divided
by 2Πε _0 x^(3). So I will tell you what I'm doing. So the final formula I had was electric field E = i
You never should settle for something that looks like that. So that's p. So p is called the dipole moment of this dipole,
and it's given by the product of the distance between the charges and the value of one of the charges, the plus charge.
So whenever I give you two charges, call it dipole, you can associate with them a vector. And the vector is, if you've got a charge
q here and a charge q there separated by a little vector r, then the dipole moment is q times the little vector r.
In our example the little vector r was 2a in the x direction. So 2a times i times q is the dipole
moment. So this means electric field is parallel to the dipole moment and falls like 1 over x^(3).
That's the most important part of the dipole. A single charge, the field falls like 1 over x^(2) if you move a distance x.
A dipole will always fall like a bigger power of x, because to go like 1 over x^(2) you've got to have net charge.
As long as the net charge is 0 the fact that there are two opposite charges that don't quite cancel each other, it always comes from the fact that the distance between them
is not 0. And the distance will appear in the numerator, and that must be the corresponding distance in the
denominator, because the formula should have the same dimensions. That's what turns the 1 over x^(2) to 1 over
this formula is good only for x bigger than a. You'll find whenever you're working with dipoles, people will always ask you to find the field very far from the
dipole. So here's a second place where I can go and find the field. That's going to be here, a, a,
and I'm going to find the field there. So that's at a distance y. So let's look at this.
That's a q here and q here. A q will repel it that way, and a q will attract it this way, and their sum will be that.
We know that these arrows have equal magnitude because this distance is the same as that distance. Therefore, it's the horizontal part that will remain.
The vertical part will cancel. You see that? We've got two arrows of the same length with this angle and
this angle equal. The horizontal part will be additive and the vertical part will be equal and opposite.
So I'm only going to compute the horizontal part. So the electric field now will be  i (that's to tell me it's in the negative x
This is (y^(2)^( ) a^(2)). That's also (y^(2)^( ) a^(2)), but I want the horizontal part of this,
Then I'm going to put on another 2 because this is going to contribute an equal horizontal part. So the E, in the end, is equal to
millimeter that's essentially the distance. If you like you can call it p divided by 4Πε _0r^(3).
Where r is the distance, if you like, from the center of dipole. Look, the point of this exercise is twofold.
One is to show you how to add vectorially the fields due to two guys. Another is, to have you understand,
at least, how to do the computation at a few simple places, where the direction of the field is not so hard to calculate.
Actually, one would like to compute it here, but it becomes quite nasty. The magnitude is not so hard, but the direction is hard to
calculate, so we'll find a shortcut. But at these two places, on the perpendicular bisector and on the axis, the mathematics is pretty
Student: Is p as a vector different from the p that you crossed out as a vector over there? Prof: No.
It's the same p. So p as a vector in our example will be the charge at either end times the distance between them.
The vector difference is 2a times i. So you can write the formula in terms of the dipole moment. Okay, so now I'm going to do the second part of the problem,
which is finding the response to E. This was all computing E. Of course there are more and more complicated examples,
but we did a few simple ones. Next is going to be, if I give you E, can you find what will happen to the charge.
So I'm going to do two examples. One is there are these two plates I mentioned to you. This one is all positively charged.
This one is negatively charged. And I shoot a particle here, with some velocity v�_o in the x direction,
and the field everywhere is down, and the electric field is some constant, j times some number E_0.
j because i is this way and j is that way. So what will this do is the question, and where will it end
What we're trying to find out is how much does it fall, and when it comes out, what's the direction of this final velocity vector.
Well, the force on this charge is equal to q times E_0 j. The acceleration will be qE_0 over
m times j in the y direction. So what'll be the position? Position will be from lecture number one of your Physics 200.
Okay, so as a function of time this tells you where the position will be. At t = 0, you are at the origin.
As t increases, it's moving horizontally due to v_0 and it's also dropping vertically due to the acceleration.
Then it's very easy from now on to compute anything you like. For example, when you want to go to that point, what will be the time?
Anybody tell me what the time will be when I go to that point? How long will it be in the region between the plates? Yep?
Student: The distance of the length of the plate divided by initial velocity. Prof: Which velocity should we take?
Student: The initial horizontal velocity. Prof: That's correct because t will be L over v_0 where
v_0�'s the magnitude of the initial velocity because x velocity is never changing. Acceleration is in the y direction.
So the time it takes to cross will be independent of the fact it's falling in the y direction. So if you put t equal to all of this you will find out
where it will end up. And that's how you make pictures on the television. You've got a bunch of plates, and then you drive charges,
and if you apply the right electric field the electron will land on a screen and make a little dot. The screen will look like this, and you're looking at it from
the other side. It'll glow. Then you want the dot to move up and down, you can move the
voltage. Then if you want to move it back and forth you've got to put another set of plates, not like this,
but coming out of the blackboard. That way you can move the electron beam in all directions. That's how you scan the television screen.
You can also use magnetic fields, but this is one simple way using electric field. All right, final thing to discuss is: what is the force of
a uniform electric field on a dipole? So let's take an electric field in the x direction like that. It's got a magnitude E_0 and it's in
So the plus charge will experience a force like that. The minus charge will experience a force like that. This will be q times E_0.
That'll be q times E_0. So dipole as a whole will not feel any net force, because the two parts are getting pulled by opposite
then of course it will drift to the right, but I'm taking uniform electric field, and because the charges are equal and opposite,
the net force on it is 0. But something is not 0. You know what something is that's not 0?
There is a torque because there is a force here and the force there. You can imagine they're trying to straighten out the dipole so
it ends up looking like this. So let's find the magnitude of that torque. Magnitude of the torque is the force times the perpendicular
You can see this makes sense. If θ was 0, if the dipole was aligned with the field, the torque vanishes,
because if the charges are like this there is no tendency to rotate. The biggest torque you get, if the charges are like this,
Put θ equal to Π by 2, you get a torque of p times E_0. That's also the reason it's called a dipole moment.
Now, I'm going to write this as a cross product, and I'm assuming you guys are familiar with the cross product, right?
You take two vectors, p and E, the cross product has a magnitude, which is p times E times sine of the angle between them,
E is like this. Turn a screwdriver from p to E, it goes into the board, and that's the torque,
and the dipole will then rotate until it lines up, or if you don't want it to rotate you've got to provide a countertorque of this magnitude.
Now, if you take any dipole and leave it, you know it will like to become horizontal, so there is a certain restoring torque that tries to rotate it
so it becomes horizontal, and it's not very different from a spring, where if you pull it from equilibrium, there is a restoring force that
brings you back to where you were. So this dipole is happiest when it's horizontal. If you go away from horizontal the torque brings it back.
So just like for a spring, if you've got a force which is kx we can assign a potential energy U, which is ½ kx^(2) so that the force
is equal to dU/dx. This is something I am assuming you guys know; the relation between potential and force.
Relation between potential and force, the potential at x_1 minus the potential at x_2 is the
This is the reason why, if you knew the potential minus the derivative gives the force, and if you knew the force, its integral will give you the change in potential.
All right, so now when you do rotations, whatever you had for force you had for torque, and whatever played the role of x is played by
θ. In other words, in rotational dynamics torque is to forcejust like one of the SAT questions.
Torque is to force as θ is to x, and work is just work. If you want to calculate the work done by the dipole,
but if you like the potential energy when it's at an angle θ and its potential energy when it's at angle 0 is the integral of the torque dθ from 0 to
dθ from 0 to θ. There's a minus sign because if θ tries to increase, the torque tries to decrease it.
That's why it has got a minus sign. Now, integral of minus sinθ is cosθ, so you will get
By comparing the two expressions you can identify U(θ) to be sorry, this is U(0) minus U(θ),
So that's the final formula you have to remember, that thecan we bring it down here? When you have a dipole in an electric field,
it has a potential energy associated with the angle which is p dot E, and if you draw a picture of that as a function of θ it goes like
this. θ = 0 is when the dipole is parallel to the field. That's when it has the minimum energy, pE.
At 90 degrees, energy is 0. At 180 degrees, it's maximum. And the torque is just dU/dθ and
you can see that the torque here and there are zero, but this is the point of stable equilibrium. That's the point of unstable equilibrium.
See, if this was a potential energy like a shape of the ground, if you left a marble there it'll stay there, but if it moved a little bit,
it would roll down hill. But if you left the marble here it'll stay there, but if you move it, it'll rattle back and forth.
That's a stable equilibrium. That's unstable. So for the dipole when it's parallel to the field you are
here, and at antiparallel to the field you are there. The difference is when you're parallel and you move it a little bit, it'll have stable oscillations,
whereas if at antiparallel if you move it a little bit it'll flip over completely and come down here. All right, so I'm going to summarize the main points so you
can carry that with you, okay? We saw today that we should think in terms of electric field from now on, and we no longer talk about direct interaction