Good morning! This is a course on electromagnetic theory, which is
aimed at EEE students; that is, to say students who are doing power electronics, machines,
power systems and other subjects. I will first discuss the syllabus that we will follow in
this course, give some details about the syllabus of the first part of the semester, discuss
the textbook and then go into some review topics, so that I can bring everyone to a
same state of readiness. So, let us first look at the syllabus of this course. The course
consists of electrostatics, magnetostatics, time-dependent fields and waves. These are the four main areas. Electrostatics introduces the electric field.
It also introduces the electric potential and it introduces an important equation namely
Poisson’s equation. So in this, this is the first part of the course. About 8-10 weeks,
we will spend in this. We will basically understand capacitance problems. The second part has
to do with the magnetic field and we will study how currents produce magnetic
fields, that is, Ampere’s law and we will also study, combining these two equations,
how particles respond to electric and magnetic fields. The third part deals with Faraday’s
law. This is the crucial part of the course for EEE students, because it is a combination
of Ampere’s law and Faraday’s law that makes all electric machines, generators, and
almost all power generation systems to work. Once we have finished Faraday’s law, we
introduce the last law, which is the generalization of Ampere’s law, called Maxwell’s equations
and from that we will derive waves. So, this is the basic syllabus we have. We
will do electrostatics, then we will do magnetostatics. We will do the first time-dependent field,
namely, Faraday’s law and then we will do Maxwell’s equations. Now inside electrostatics
itself, there are several topics and I plan to cover them as follows. First we will do
review and introduce vectors, that will take probably two lectures. Then I will introduce
Coulomb’s law and the electric field and this seems a little excessive but I plan to
take around two lectures to pin down all the concepts.
Once we have done that, I will introduce Divergence and then Gauss’s law and that should take
two more lectures. Once we have done Gauss’s law, we should be able to go from electric
field to electrostatic potential. And from electrostatic potential we should get to Poisson’s
equation in another two lectures. Once we have finished these fundamental ideas,
then we will start applying them. First of all, what is the notion of capacitance and
stored energy? I will talk about dielectrics and I will introduce at this point, an important
idea which will keep coming back, which is, Boundary Conditions, again two lectures, and
finally solution of Laplace’s and Poisson’s equation and application to capacitors, two
lectures. So the first part of our course is going to
be basically like this. So you can see that its going to go rather slowly. 12 lectures
I plan to take. I may compress one or two parts and expand one or two parts but this
is more or less the speed at which I will teach the material. After that we will move
on to magnetostatics and after that, introduce Faraday’s law and finally finish up with
electromagnetic waves. Okay. So as I said I first would like to start
with a little bit of review. Before that let me also introduce the textbook for the course.
The textbook is this book. It is called “Engineering Electromagnetics”. The authors are W. H.
Hayt, Jr and John. A. Buck. Professor Hayt is one of the most famous authors
of Electrical Engineering books. Hayt and Kimberly and many other books that he has
been an author of are now classics. Unfortunately he passed away quite some years ago and his
books have been continued by John Buck. This book is a very elementary book. There are
more advanced books available but I think it is also the best introductory textbook
for Electromagnetics. Anyone who wants to go further into the subject, there are many
excellent books and I will mention them as I mention various topics but as a self-contained
textbook this is one of the finest books around. Okay. Now one of the problems that students
have with Electromagnetics is that they have already learnt it in school. They have already
learnt it in Physics. So they do not know what is left to learn and they are scared
actually to find out what they have left to learn. So let me review what you already know
so that we can put a context to this course. From your first course in electricity and
magnetism you already know that there is something called Coulomb’s law. You know that if you
have two charges, q 1 and q 2, and if these charges are separated by a distance d, that
there is a force that is proportional to the amount of charge q 1, the amount of charge
q 2 but is inversely proportional to the square of the distance. That is, the further apart
these charges are the less they exert force on each other. You also know that the smallest
charge you can find is the charge of the electron and the proton. The proton is positive, the
electron is negative. The amount of charge that an electron or a proton has is 1.6 into
10 to the minus 19 Coulombs. The unit of charge has been named after Professor Coulomb himself. Okay now you also know quite a few other things
about electricity and magnetism which you have learnt in school, you have learnt in
college. One of the things you know is that, if you have currents, if you have a wire
and the wire is carrying current, then a magnetic field develops and the magnetic field goes
around this wire. It doesn’t radiate away from the wire. It circles the wire. And there
is a rule for how it circles the wire. You are supposed to take your right hand, point
your thumb along the current and the direction in which your fingers will curl will tell
you the direction in which the magnetic field will go around, all right? Now there is another thing that you know,
which is, if you have a magnetic field like this and you put a proton, this proton will
feel the force, if it is moving. So if you have this proton, it’s got a velocity v,
this proton will feel a force, in fact, the general force law that you have is, if there
is a force, equals the charge times the electric field plus the charge times the v cross B
of the magnetic field. When I put lines underneath these variables
I mean they are vectors. This is just one kind of notation. The other kind of notation
you could have written is, q E with an arrow above, plus q v, with an arrow above, cross
B, with an arrow above. Both are used and both are all right. Since I am comfortable
with this notation, and it is easier to draw, so I will be using it. But they mean the same
thing, alright? Okay now what else do we know about electricity
and magnetism? We know that there are things called capacitors. You can go out to the electronics
shop and you can buy them. These capacitors are made of silver foil and are separated
by some insulator. Symbolically we draw them like this. So you have aluminum-silver-copper
plates which are separated by a very small distance. When these plates are brought very
close to each other and then you apply a charge, you put a charge on the top plate and a minus
charge in the bottom plate, you find that there is stored energy inside here and this
leads to a concept we call Capacitance. And you know that the capacitance of a plate capacitor
like this is equal to epsilon A by d where d is this distance and A is the area. So this
is something you would have learnt in school. Now similarly you have learnt of magnetic
stored energy. Magnetic stored energy comes in the form of inductors they are crucially
important for machines because all machines in generators are basically inductors. Transformers
they are all inductors. So what is an inductor? An inductor is a coil
in which current is flowing. This current, as it flows, creates magnetic field, and as
we will show later, using these same arguments here, the inductance the current in the coil
creates a magnetic field that threads all these coils all these turns. This magnetic
field represents stored energy. Again it is a different kind of stored energy called magnetic
stored energy. Okay so the these are the kinds of concepts
we have already met. We know of Coulomb’s law. We know that currents can create magnetic
fields. We know magnetic fields and electric fields can create force. We know that if you
bring plates very close to each other you can create something called Capacitors which
store electric energy. If you twist turns of wire very close to each other you can create
magnetic fields which are stored magnetic energy. So these are all the concepts you already
know. So what is there left to learn? Basically in this course what we are going to learn
is how to develop flexible ways of thinking about electric and magnetic fields and how
to develop powerful techniques to analyze these electric and magnetic fields. Supposing the geometry of this capacitor was
not simple, what would we do? Well we have techniques for calculating capacitance of
anything. So that kind of topic is what we will be attacking in this course. What are
the tools that you will require? Clearly, electromagnetic theory is a very mathematical
subject. The mathematical subjects that you will need
to be good reasonably good at include vectors- we will review vectors- but still you should
have learnt them. You need to understand differential equations. More than vectors you need to understand
what vectors can do in the presence of differential equations, so this is called vector calculus
and you need a different form of differential
equations called partial differential equations. Up to about fifty years ago that’s all you
required but since the advent of computers the entire field of electromagnetics has completely
changed and a new tool has appeared and this new tool is now universally used in this subject
and this is the tool of computer modeling and simulation. So if you look at the four topics, you need
to understand what vectors are, you need to understand vector calculus. We will be covering
this area subject in some detail in this course, the textbook does quite a bit of treatment
on it. Differential equations, you should have learnt by now. We will basically assume
that you know how to solve differential equations. Partial differential equations- well, the
techniques for solving them we will learn in this course, but you will have to have
already learnt some amount or else you will find this part rather difficult. As far as
computer modeling and simulation is concerned the techniques that are required will be introduced
in the course itself. Okay so that’s as far as what the course is going to be, what
the course requires and what we plan to cover in the first twelve weeks. Now for the rest of the lecture I am going
to talk about some other preliminaries, namely, vector fields. Okay so when we talk about electricity and
magnetism we deal with certain kinds of quantities, for example, we say that if a charge Q is
present and another charge q is also present then wherever we keep this small charge q
there is a force on it. If the charge q is kept here, the force is in this direction.
But if I took the same charge and put it down here the force is in this direction. That’s
what Coulomb’s law tells us. But we can pretend even before the charge was placed
here the force was waiting to happen. So that when the charge is placed the force is actually
felt. The concept of a force defined by every point in space is called a force field. So
force field or a field in general is a quantity which is defined for all positions. So a quantity like q 1 is not a field, q 1
is a single value, q 2 is again a single value, but when we talk about variable distance d,
this force which is a function of d- d is a variable- that force becomes a field. So
field is a quantity and is a physical quantity that takes on different values at different
locations. So the commonest example is the force field. There are plenty of other examples.
For example, supposing this was the earth and there is an object suspended above the
earth, this object feels a force. The force is the force due to the gravity. Now there
is a there is a gravitational field, namely, a force waiting to happen, which is surrounding
the earth. In all directions it is trying to pull things towards the earth. When the
object is here it gets pulled down this way. When the object comes here it gets pulled
down this way. If the object comes all the way here it gets pulled down this way. So this is another field and it is called
gravitational field. Similarly you can have a field which is simply the location of a
point. Every point has coordinates x and y and if it has height it has a coordinate z.
Now if we define some point, let’s say the centre of the earth, as our origin, then we
can draw a line from the centre of the earth to any point and that point would have an
x, a y, and a z. Now supposing we attach to x, the direction
along x, and I denote it this way, I put a little carat on top of the quantity. What
I mean by that is, take this direction, give it magnitude one so it has no magnitude, its
magnitude is just one but it has direction. The magnitude is sitting inside x. The direction
is sitting in x carat. Similarly, magnitude inside y, direction inside y carat. Magnitude
inside z, direction inside z carat. This defines a particular direction. It is
the distance from the centre of the earth to wherever we are measuring. Now this too,
is a vector field because it depends on where you are measuring. You go to another point,
that point has a different value for x, y and z. And because it has a different value
for x, y and z, it is a different vector. So this too is a vector field. Not only can
you have vector field, you can also have scalar fields. For example, if you look at the newspaper
it will show you the map of India and it will tell you what the temperature is at different
points in the country. That map showing you isotherms saying that this is thirty four
degrees, thirty two degrees, this is thirty degrees, this is actually defining a scalar
field. It is saying that at every point in this subcontinent, you can associate a temperature.
The scalar field is the temperature. At this point, the temperature is thirty four degrees.
At this point, the temperature is thirty two degrees and at some point on this line the
temperature is thirty degrees. Like temperature, you can have many other
kinds of scalar fields, for example, pressure; temperature itself, and a well known scalar
field that we will encounter, which is electrostatic potential. These are all scalar fields. The
difference between scalar fields and vector fields is only one thing. Scalar fields have
magnitude. They have a number but they don’t have a direction. Vector fields not only have
a number they also have a direction. The all of these- whether it is scalar fields or vector
fields- the quantities we are looking at, we are describing, must be physical quantities.
So we cannot say that the colour of a car is going to be a scalar field. That is a very
arbitrary kind of quantity, so it doesn’t come under this kind of description. Okay
so if it is a physical quantity we are describing, that qualifies to be a field, and we can write
equations for it, and we can solve those equations and predict what these fields will do. Now I have started talking about vector fields
but I have not said what a vector is. That’s because you people are supposed to have learnt
all about vectors earlier, but let me revise. We can draw coordinate system, that is, x,
y, z and a vector is something that has a magnitude, that is the length and a direction
that is, it makes angles with all three axes. So a vector is an arrow. The length of the
arrow gives you the magnitude, the number associated with the vector; the direction
in which arrow is pointing determines its direction. But that is all we can do with vectors. They
are not of any use. Most vectors have certain operations you can do with them. For example,
vectors can be added. How do you add a vector? We start with the first vector and place the
second vector with its foot on the nose of the first vector and then join the tail of
the first vector with the nose of the second vector. This dashed line is defined as the
sum of this vector and this vector. Now in Mathematics, whenever you define operations
you like to classify them. And your textbook classifies them quite nicely. This way of
defining addition; it is commutative, it is associative, it has an identity, you can read
all that in your first chapter. But for us, it is enough to know that we can place one
vector on the top of other vector or we can place the other vector on top of the first.
Since this is a parallelogram whichever way you do it, it will work the same. Vectors have another property, which is, they
can be scaled. See, if I have a vector, I can also define twice the vector- which is
a line that’s twice as long- which is a line as twice as long, pointing in the same
direction. So I want three times- I draw a line that is three times as long, pointing
in the same direction. So vectors can be scaled and they can be added. If you take these two
properties and you combine them, you get something which is called a mathematically which is
called a field. Mathematical properties of a field are written in your book. It is not
an accident that we call the electric field a field. The electric field satisfies certain
mathematical properties. These mathematical properties are basically that the vector addition
behaves itself as we expect it to and the vector scaling also behaves itself pretty
much as we expect it to. They are all common sense properties so I won’t go in what those
properties are. But now having defined these things called vectors what are the operations
you can do on them, besides adding and scaling? The two important operations you can do which
are related to physical quantities are dot product
and cross product. I will repeat this next time, but let me introduce these two concepts
so you can think about them. Supposing a ball is rolling down a hill the
force of gravity is downwards at every point. Now the ball when it rolls, it is not going
straight down, it’s going sideways. And we know that, if this were frictionless, that
the kinetic energy of the ball, both in if you add up both the rotational energy and
the falling energy, the kinetic energy of the ball, when it reaches here, is related
to the loss of potential energy. So we know that increase in kinetic energy is equal to
m g h. But the ball would have gained the same energy had it fallen straight down also. So it seems that how it fell to this point
doesn’t matter, all that matters is how much it fell. So in other words if the ball
fell from this point to this point, straight down, it gained m g h. If it fell like this,
it still gained only m g h. The force due to gravity is uniform and downwards. So somehow you need to get a number that is
constant, regardless of which way the ball is falling, whether the ball is falling straight
or the ball is falling sideways. So the answer to that is we look at this angle.
If you look at that angle theta this length, h you can write a relation. You can say that
h is equal to l cos theta. Cos theta is, the base divided by the hypotenuse. And so h must
be equal to l cos theta. So what that means is that, if you are moving in any direction,
this is the direction you are moving, and this is the direction of force, the amount
of energy you gain is not equal to force times the direction distance along the direction,
rather it has to do with force times the direction times cos theta. That’s what you get here
that is instead of m g l, you get m g h, substituting for h you get m g into l into cos theta. In vector notation the same idea can be written
down. The idea is you have the vector m g. I have put a line underneath the g to indicate
it is now a vector. This 1 can be given a direction as well, l and then this is cos
theta and we take care of cos theta by putting a dot. This dot implies taking the angle between
the two vectors finding the cosine and multiplying the magnitudes. This operation is extremely
important. It is probably the most important vector operation you know. It is very important
in all operations we do regarding work, regarding kinetic energy, potential energy and you would
have encountered in many places. Now let’s take a look at another operation
which is the brother of the dot product. When a magnetic field is present
and you have a proton, and the proton is moving in a certain direction, you find that the
proton in the presence of the magnetic field B does not move either, does not change its
direction either in the direction B or in the direction v. In fact the proton tries
to change its direction in the third direction. It tries to go ninety degrees to B as well
as ninety degrees to v. And what is all the more, the amount by which this force happens,
this force is proportional to the magnitude, the number associated with v, the length of
B into the sine of this angle. Now in the dot product- the magnitude multiplied
by cos theta, in the cross product magnitudes are multiplied by sine theta. But the most
important thing of all is the direction that B cross v has. Let me summarize: dot product
is a scalar, it has only magnitude- no direction and it has the vector dot another vector is
the magnitude of the first vector magnitude of the second vector times cos theta. The
cross product, it’s a vector itself. So cross product not only has a number associated
with it, it also has a direction associated with it. So if I have a vector v and I find
its cross product with a vector w, its value is the length of the vector v, the length
of the vector w, times sine theta. And its direction is in the third direction namely
normal to v and w. This is a very important place where this keeps appearing. Supposing we have two vectors which are at
right angles. Then we know that the area of this square is this vector magnitude of the
first vector multiplied by magnitude of the second vector. This, this is just ‘multiplied
by’. Area of the rectangle is product of the two sides, but supposing you have two
vectors which are not at ninety degrees to each other then we know that we have to draw
a parallelogram and the parallelogram has an area that’s equal to the rectangle with
the same base. So it is this distance which is magnitude. I will call it v. This distance
is the magnitude w, but this height is magnitude w sine theta. So the area of this parallelogram- it is same
as the area of the rectangle- is magnitude v magnitude w sine theta. And this is the
same formula that you get for cross product. You can see that cross product deals with
sine theta. So the cross product is actually very closely related to finding the area of
rectangular objects when you have two vectors which are not at ninety degrees with each
other the area is this and when you try to give a direction to an area the only natural
direction they give is out of the page. You cannot give the direction of an area as this
way or this way. If there is any one direction that defines an area, it is the direction
that is pointing away from the area. It’s a very strange way of thinking about things
but it has proved to be very, very useful in physics. So if you now take, say a square. A square
has some hidden sides also. This area will point outwards like this, this area will point
outwards this way, this area will point upwards, that is, areas point away from their surfaces.
These areas are also called normals and the direction of a cross product is nothing
but the direction of the normal. So if you take any two vectors v and w, the idea of
the cross product is, make the parallelogram that is constructed by v and w and then point
in the direction of its normal. When you do that you have the direction of the cross product
and the area of this parallelogram is the magnitude of the cross product. As I said it’s a very strange kind of definition
but, in fact, not only it is useful, half the equations of electricity and magnetism
use the cross product. They use a complication of the cross product but are basically the
cross product. Where does the cross product come? Well the first place where it comes
is in Ampere’s law where you have a current I and you want the magnetic field B. When you want the magnetic field B from a
current, you would like to use Ampere’s law. Ampere’s law tells us that the line
joining to this wire to B cross this current has something to do with the magnetic field
magnitude and direction. The actual equation involves a differential equation, so I won’t
put it down right now, we will develop it as we need to. Similarly if you look at the
force on a charge- we have already talked about that- it is q v cross B. So if a proton
moves in a magnetic field, it feels the force that is perpendicular, that is normal to both
v and to B. And so you find that if you have a magnetic field pointing in the z direction
and the [electron] proton is moving in the x-y plane, the proton is at all times feeling
a force that is again in the x-y plane but ninety degrees to the velocity. So what the
proton does, it goes in a circle. I will come back to this next time. This is
basically one of the most common examples of the cross product.
