In the last lecture we studied the basics
of vectors, we defined the basic operations
on vector like cross and dot product and then
we saw the differential operations on vector.
So we introduce this differential parameters
which is vector operator called del which
is defined as this and by using this operator
then we defined three operations on vectors
the del of f where f is a scalar of quantity
and del of f gives you what is called gradient
of the scalar quantity f so this is del of
f is a vector quantity and physically we saw
it tells you the maximum rate of change of
this function f in the three dimensional space.
Then we defined for the vector fields the
operations which was the operation del dot
f and this quantity we call the divergence
of vector f, this quantity is a scalar quantity.
Thirdly, we had defined the vector operations
which was the cross operation and we called
that as the curl of vector f so del cross
f is given as the determinant of the matrix.
In this lecture today we will try to get a
physical feel for this operation: the divergence
and the curl operations and we define then
the basic theorems which are used in the vector
operation. So just to get a feel for the dot
product; essentially if to you consider the
vector field like a fluid consider a small
box or small volume and ask how much is the
net flow of fluid from that box per unit volume
that quantity is nothing but the divergence
of vector.
So if you imagine the vector field something
like a fluid then the net flow which is coming
out of the volume per unit volume will be
essentially represented by the this which
is the divergence of the vectors. Similarly
when we look at the curl of the vector as
the name suggests it is something curling
or there is some rotation involved here.
If you consider the vector field and keep
an object in treat the vector field like let
us say again like a fluid flow or some surface
of a river and try to keep some object on
the surface of a river because of the differential
flow of the layers of the water there will
be some kind of a rotational effect which
will be created on the surface of the river
that is the effect which is captured by this
operator what is called the curl operator.
So if you define net rotation created on an
object per unit area of the object then that
quantity essentially is the curl of that vector.
So just to get a little better feel let us
ask what kind of fields would give me divergence
and what kind of field will give me the curl.
So if I draw let us say a vector field which
is given by that again we are writing the
arrows which are representing the vectors,
so if I consider a vector field like this;
if I consider the magnitude of the vector
here here here they are same, it is here here
here is same, here here here is same but if
move in this direction this value this value
is same but this value is different, this
value different and so on.
Now imagine if I keep a small area here then
depending upon the value of the vector it
will create some kind of a shear on this object
so this vector will try to rotate the object
this way whereas this vector will try to rotate
the object in the anti-clockwise direction
so there is a net rotation effect which is
created on this object. So if I have a vector
field something like this then I will have
a rotation created and this vector field will
have a curl in this region. If I consider
a field which is like that 
that means now the field magnitude is changing
in this direction so here certain value of
the vector the vector increases, increases
and so on.
If I consider now a small volume in this region
and if I treat this vector like a fluid it
will represent... the fluid which is going
inside is having the value which is this whereas
the field which is coming out of this will
be having a magnitude which is much larger
so then it will be give me a net flow of fluid
it form this volume if I keep some volume
at this point.
So, if I have field something like this then
it will have a divergence; if I have something
like this then it will have a curl. So whenever
we have some kind of a flux coming out of
a volume then the concept of divergence is
the correct concept it can capture that effect
of something oozing out of that volume or
getting inside that volume.
If I have a physical phenomena where some
rotational kind of effects are involved then
the phenomena which can captured that effect
is a curl.
So divergence curl essentially are the mathematical
operations which capture this phenomena. So
you will see later on when we go to the general
analysis electric and magnetic fields. Essentially
the concept of physics are mathematically
captured by the curl and divergence operations.
Another operation which is defined in terms
of the del or this differential operator is
what is called Laplacian operator.
This is actually a second-order differential
operator and this is denoted by del square
which is equivalent to del dot del. So if
I treat the del like a vector then the dot
product of this del vector is the operator
del square. So this operator the Laplacian
operator is a scalar operator and therefore
this is... if I take a dot product of the
del operator and if I treat this like a vector
I will get the dot product of del with itself
so this will become d2 by dx square so this
is d2 by dx square plus d2 by dy square plus
d2 by dz square.
So the Laplacian operator is a second-order
differential operator and this operator is
a scalar operator. If I operate this on a
scalar quantity the Laplacian operator will
be del dot del of the scalar quantity. So,
for a scalar function f 
function f the del square operator del square
of f will be equal to del dot del of f. Since
we have defined this quantity del of f as
the gradient of the scalar function f and
this dot represents the dot product or the
divergence the Laplacian of the scalar quantity
is nothing but a divergence of the gradient
of the scalar function. So this Laplacian
operator in this case is divergence of gradient
of the scalar function.
The del square operator however is not restricted
to the scalar function only. The del square
operator can operate also on the vector. And
if I follow the same thing as you have done
here that del dot del and if I put a vector
in front of it I will get del dot del of the
vector f capital f but that will not have
any meaning because you know it defines the
operation del of the vector quantity. When
the del operates on a vector quantity it can
operate either a divergence or the curl. So
in this case when you operate this one on
the vector quantity we directly take this
and operate this directly on the vector quantities.
So, when you have a Laplacian of a vector
f then del square of f where f is a vector
quantity that is equal. Operating this on
all the three components of vectors so essentially
you take the second derivative d2 by dx square
of the entire vector which is three components;
you take second derivative with respect to
to y of the entire vector and you take second
derivative with respect to z for the entire
vector.
So in this case although the del square operator
is a scalar quantity that operator is a scalar
operator when it operates on a vector you
get a quantity which will be a vector quantity.
So, when del square operates on the scalar
quantity you get a scalar function. When you
operate this this on a vector quantity you
get a vector quantity. So in this case you
will get d2 by dx square f plus d2 by dy square
f plus d2 by dz square f.
What it means is if I expand f in its components
f x f y f z you have d2 by dx square for all
the three components, d2 by dy square for
all three components and d2 by dz square for
all the three components. So if I expand this
and if I say the f is f of x into x plus f
of y y plus f of z z and I can substitute
into this so del square of f will be equal
to d2 by dx square into f of x x plus f of
y y plus f of z z plus d2 by dy square same
thing plus and so on. Then I can combine the
next component in all the components. So,
that will give the d2 f x by d x square d2
f x by dy square d2 f x by dz square that
will be the ex-component of this vector. So
this will be d2 f x by dx square plus d2 f
x by dy square plus d2 f x by dz square that
will be the ex-component plus similarly for
y component and so on.
So essentially I take this Laplacian operator
which is operating on each of the components.
So this is I can say this is equivalent to
saying this this is del square of f x into
x plus del square f y into y plus del square
f z into z.
So if I take each of the components of the
vector make the scalar Laplacian operation
on each of the components and write the final
vector that will be the quantity which will
be del square f where f is the vector quantity.
So later on when we do the analysis of the
vector fields we will see this operation will
be needed for solving the electromagnetic
problems.
These are the differential operators and then
you have to do the operations which are the
integration of the vector fields so we require
certain operations which we can do in the
integration kind of operation. So if I have
a vector field there is a possibility that
I can take the integral of this vector field
in a plane along a contour or along a line
so if possible I have certain vector fields
and I can take the integration around a path
a contour. If possible I can take the integration
of this vector on a surface which could be
a closed surface or open surface so I can
have something like that is a surface; I will
require the vector field which is something
like this, here again vector field could be
like that so I will require the integration
of this vector along this path, I may get
the integration of this vector on this surface
or I can take the integration over the volume
if is the surface is a closed surface.
So now when we do the integration if the integration
is done along this path we call this as the
contour integration; if I take the integration
over a surface we call that as surface integration
and since surface is a two dimensional thing
essentially we will have a double integral
of the surface integration; the contour integration
is a single integration and if I go to a volume
integration then it will be a triple integral
in the three dimensional space.
Then we have an important theorem which connects
the line integral to the surface integral
and surface integral to the volume integral.
So if I have a volume here I will have integration
over the volume, so I have here line integral,
I have surface integral and I have volume
integral and invariably we need to change
from one integration to another; we may have
to convert from the line integral to the surface
integral or from surface integral to the volume
integral. It should be kept in mind, however,
that the integral is a scalar quantity. So
when we do the line integration the final
answer will be a scalar quantity, when we
do the surface integration the final answer
will be a scalar quantity.
So now we can define the surface integration
and the volume integration and the line integration
that is if I take some vector A some vector
field and say denote it by A is my vector
field then the line integration around a path
of contour some contour c is integral A dot
dℓ where dℓ is the segment along the length.
So this quantity is dℓ and this is the direction
of segment; if I am integrating along the
contour like this so dℓ is a vector quantity
because it has a length and it has a direction.
If the contour is closed contour then I will
have the integral which will be a closed integral.
So here this is open integral, open contour.
If the contour is closed then integral will
be denoted by this: A dot dℓ this is closed
contour.
So, at every location essentially what we
are doing is we have this vector A, we find
out the dot product of A and dℓ at every
location, you add up that along the contour
and that is this line integral. So this is
the line integral. We can do the similar operation
for the surface integration.
So let us say we have some surface like this
and there is an infinitesimal area on this
which is given by da, this is the direction
of the area this is the unit vector in the
direction perpendicular to that infinitesimally
small area and the vector field is again A
so I can define the surface integration; again
the surface may be closed, it may be open
so here this vector and the area which is
electrical quantity so this is infinitesimally
small area and the direction of the area is
the direction which is perpendicular to this
area so da vector area is the infinitesimally
small area and then multiplied by the unit
vector which is in the direction perpendicular
to this area. So, if I take the dot product
of these two A and da and if I sum up over
the entire surface, that gives me what is
called the surface integral.
So we have here now surface integral. So,
if I have a surface which is an open surface
then I have this double integral 
A dot da this is for open surface and A dot
da for the closed surface. So the integration
now is a double integration because we are
talking about a surface and if you write the
integration like this then it represents the
open surface, if you write loop around the
integration then that represents integration
around the closed surface. Again you can note
here we are taking a dot product of the vector
A and the area at that location. So this quantity
is a scalar quantity so this integration is
going to be a scalar quantity.
Now, direction of the normal which we take
for A if it is a closed surface then the direction
of normal generally is taken as the outward's
normal from that volume that direction is
taken as the positive unite direction. However,
if you consider a surface which is an open
surface then of course there is no preference
for defining the unit normal so we can define
the unit normal in either direction.
If I consider a surface like this the unit
normal can be defined as this way what can
be defined either this way. If I am having
a surface which is a closed surface like this
then the normal which is coming out from this
surface that normal is taken as the positive
direction. So, while defining the unit normal
for the surface there is no specific notation.
However, later on when we will try to connect
the contours to the surface for an open surface
that time we will follow again the convention
of the right hand rule and then we will specifically
choose the direction for the unit normal for
the surface in the contour.
So in this case let us say we can define if
your volume was closed, surface was closed
then we define the unit normal coming outwards
if the surface is open then any direction
can be taken as the direction of the unit
normal.
The third possibility we said is the volume
integral. So since we are now defining this
quantity over the volume we have some function
which is a scalar quantity which is filling
this volume. See if I integrate the total
scalar quantity over this volume that gives
me the volume integral. So I have some volume
here and I have some scalar quantity filling
this volume which is given by f. if I take
an infinitesimally small volume in this that
volume let us say is given by dv where v is
the volume then integration over this will
be a triple integration so we have here volume
integral which is triple integral; this function
f dv.
So, essentially when we try to now relate
the different integrations: the line integration
to the surface integration to the volume integration
essentially the vector fields first are operated
by the operator del and then there is the
relationship between these del operated fields
in the integration domain. So there are two
important theorems which essentially relate
this and they are called divergence theorem
and the Stokes' theorem. So we have an important
theorem what is called a divergence theorem
which converts a surface integral for a vector
field to a volume integral.
So if I have a vector field A vector field
A then the divergence theorem states that
for a closed surface the integral A dot da
that is equal to the volume integral of divergence
of A. So if I consider a surface it has a
total surface area let us say is given by
s and the total volume of this which is v.
If I take a surface integral of the vector
A on the total surface S then that is equal
to the volume integral of the divergence of
A over this volume.
So if you are having a vector and if you know
the surface integral of this vector the surface
integral can be converted into a volume integral
by operating this vector A with a del operator
with a dot product that means it makes a divergence
of vector A and take the volume integral of
that that is same and the surface integral
of the vector A. This theorem is called the
divergence theorem. So whenever we have a
need to convert from volume integral to surface
integral or vice versa this theorem comes
very handy.
Another theorem which is again an important
theorem and that converts the surface integral
into the line integral and that theorem is
called the Stokes' theorem.
If I have an open surface now something like
this I have a contour of this surface so this
is the contour c and I have this surface so
I have some surface area s. So the surface
is open and the boundary of this surface is
the contour c. Now I can have the unit area
here and that is the direction of let us say
n which is the unit vector for the area, then
I have this contour for which I have to define
the direction of ℓ; when I can do integration
I will require ℓ so essentially first you
have to say how should I define this ℓ with
respect to n and the convention is that if
I again follow the right hand rule if my contour
goes like this then will be like this so the
n will be coming inward. If I take contour
which will be like this then the n will be
going outwards.
So, if I take the direction of the contour
like this that means the dℓ vector if I
define that way so if I take the contour positive
direction like this then the direction of
the unit vector for the surface will be like
that. If I change the direction of the contour
integration in the opposite direction the
normal direction of the surface integral will
be inwards. So, for this then we can write
the contour is a closed contour in this case
so the stokes theorem states that over this
contour c the line integral of this vector
A you have a vector field again here which
is A that is equal to the surface integral
and in this case it is open surface curl of
A dot d.
So the line integral of this vector a on this
contour is equal to the surface integral of
the curl of this vector A. So this theorem
can be used now for converting the line integral
to surface integral and vice versa. So again
summarizing this if I have a closed surface
then the surface integral and the volume integral
can be related by the divergence theorem and
if I have an open surface then the line integral
and the surface integral can be related to
the Stokes' theorem. So, later on for solving
the problem of electromagnetics especially
when we write the Maxwell's equation for the
integral and differential form these two theorems
become very important, become very handy in
converting from the differential to the integral
form.
Having understood these basics of the vectors
now we can go to the basic quantities of the
electromagnetics which we are going to make
use of in further analysis that is the quantity
like electric and magnetic fields when the
origin of the entire electromagnetic phenomena
is the basic charge. See if I consider charge
you know from our basic Coulomb's law this
charge has the effect around it so if I put
another charge in the vicinity of that it
experiences a force. This effect is measured
by a quantity what is called the electric
field.
So, if I take a charge and go into the vicinity
of that I experience a force which is characterized
by a quantity called electrical field. However,
if I keep this charge in motion then it constitutes
a current because current is nothing but the
rate of change of charge so if the charge
starts moving you have a sustained flow of
charges that simply gives you current and
then we have the magnetic fields. So the same
charge well it is static stationary it gives
you the electric field and when it starts
moving then it gives you current and that
gives you the magnetic field.
The charge gets accelerated also. So if you
accelerate the charges then it gives you the
electric and magnetic fields both. So essentially
we are dealing with the quantities here: the
charges, currents, electric and magnetic fields
and try to establish the relationship between
these quantities. The relationship which we
have between these quantities is given by
what is called Maxwell's equation.
So essentially now starting with the basics
of these quantities we will try to establish
the relationship between them from the laws
of physics and when we write the laws of physics
in the mathematical form using the vector
notation and the vector theorems which you
establish we get what is called the Maxwell's
equation. So the quantity which we have now
is what is called first quantity is the electric
field E which is nothing but the force experienced
by unit charge.
So if I have some charge here by Coulomb's
law I will experience a force at the location
in the vicinity of this charge. If I measure
this force per unit charge that quantity is
what is called the electric field at that
location. Since here we are talking about
the force per unit charge, the electric field,
the vector quantity so it has the directions
and it has the magnitude. The unit of electric
field is volts per meter. So electric field
is given by volts per meter. then we have
a medium property that if I measure the electric
field in let us say vacuum I will get certain
force, if I measure the same thing if I change
the medium parameter to some other direction
then the force will change so the quantity
which does not depend upon the medium parameters
is the electric displacement vector.
So the electric field is a quantity which
is related to a charge which is producing
this field. But it also is related to the
medium parameter which is what is called the
permittivity of the medium. So we have a medium
parameter called permittivity which is denoted
by epsilon and it has a unit: Farad or meter.
So if the medium parameter changes the permittivity
of the medium changes and because of that
the electric field measured at that point
changes. the permittivity of the vacuumed
or the free space is denoted by epsilon 0,
so for the free space we have epsilon is equal
to epsilon 0 and its value is approximately
1 over 36 pi into 10 to the power minus 9
Farad per meter.
So, in the space which is nothing which is
vacuum; even for that the permittivity has
a value and which is... many times this is
also written as 8.86 10 to power of the minus
12 and so on. However, it is easy to remember
1 upon 36 pi into 10 to the power minus 9
rather than remembering it as 0.8 something.
Generally we prefer to write the permittivity
of the free space as 1 over 36 pi into 10
to the power minus 9 Farad per meter.
Then if you take another media whose permittivity
is some epsilon, the ratio of epsilon to the
epsilon 0 is what is called the dielectric
constant or the relative permittivity of the
media. So we have the dielectric constant
epsilon r also called as relative permittivity
that is equal to the permittivity of the medium
divided by the permittivity of the free space
that is epsilon 0. Then the quantity which
is independent of the medium parameters that
is what is called the electric displacement
vector that is the product of electric field
and the permittivity of the media.
So we have a quantity electric displacement
vector 
denoted by d which is equal to permittivity
of the media multiplied by the electric. So,
when the medium property change the epsilon
changes but the d remain same; d does not
depend upon the medium properties it depends
upon the actually charge which is creating
this field irrespective of which media it
is creating this field. So d remains same,
it does not change from media to media, so
the quantity which changes is the electric
field depending upon what is the permittivity
of the medium.
Now this quantity epsilon in a general media
can be constant everywhere, can be uniform
or it can vary as a function of the space
in three dimensional space. It can also depend
upon the direction. What that means is if
I measure the permittivity in certain direction
it has certain value, if I measure the permittivity
in some other direction it will have another
value, so in general this quantity epsilon
may be direction dependent, it may be space
dependent so if epsilon varies as a function
of space then we call the medium as a inhomogeneous
medium.
So epsilon varies with the space, this gives
you the inhomogeneous medium. If epsilon is
a function of direction, if epsilon is direction
dependent then the media is called anisotropic
media. So if epsilon varies as a function
of space we call the medium inhomogeneous,
if the epsilon is direction dependent then
the media is called anisotropic, if the media
is not varying as a function of space then
we call that medium homogeneous and if epsilon
is not varying as a function of direction
then we call that media as anisotropic media.
In this course we deal with the media which
are homogeneous and isotropic media. So here
we consider only homogeneous and isotropic
media. What that means is that the dielectric
constant of the medium or permittivity of
the media is neither direction dependent nor
it is varying as a function of space. In general,
however, if the media was an isotropic then
this quantity epsilon is not a scalar quantity.
In fact it becomes a 3 by 3 matrix and D is
now equal to this 3 by 3 matrix epsilon multiplied
by this E which is the vector.
So, for anisotropic case the epsilon is a
3 by 3 matrix whereas if I take a medium as
isotropic then epsilon is a scalar quantity.
So, in this course we essentially deal with
the media for which epsilon or the dielectric
constant is a scalar quantity. What that means
is that in a homogeneous medium if this epsilon
is a scalar quantity that D is nothing but
a scaled version of E. So if I compare D and
E they have different magnitude these two
vectors but the direction of E and D are same
that means in anisotropic medium the displacement
vector and the electric field they are in
the same direction. However, if I take the
medium which is isotropic that in general
this is a 3 by 4 matrix so essentially it
rotates this vector E and in general the displacement
vector and the electric field vector are not
in the same direction.
So we say that this is E and this D this is
E, this will be the case if the medium is
isotropic whereas this will be the case if
the medium is anisotropic because we said
that the displacement vector D is given in
this case by 3 by 3 matrix so let me just
write that as epsilon x x epsilon x y epsilon
x z epsilon y x epsilon y y epsilon y z epsilon
z x epsilon z y epsilon z z multiplied by
this vector which is Ex Ey Ez. So this is
equal to three components of the vectors D
x D y D z.
So you can see here; if I had a vector whose
components are E x E y E z after transforming
to this matrix which is the permittivity of
the anisotropic medium that will give me displacement
vector which will not have the same direction
of the electric field so I will have a situation
as something like that. Then we have a quantity
which is useful which we define and that is
the electric potential at a point in the field
and that is nothing but the negative gradient
of the potential is the electric field. So
the electric field is related to the electric
potential; electric potential is the scalar
quantity V and that is related to the electric
field as the electric field is minus gradient
of the voltage.
So if I know the potential at a point then
I can take the gradient of that and that gives
me the electric field at that location. From
here the unit which we have got for the electric
field essentially comes from this definition.
But here we have defined the potential which
is which is like that, the unit for the voltage
is volts, the del operator is a differential
special differential operator so it is d by
dx so its units are 1 by length or per meter
so that gives me the unit of the electric
field which is volts per meters.
So this relation for converting or for finding
out the electric field from the potential
it turns out very handy; whenever we find
try to find the electric field in a general
complex distribution of the charges, if you
calculate the electric field at a particular
location and if I find the electric field
because of each component of the charges which
are distributed in space you have to carry
out the vector additions at that point for
the electric field so generally it turns out
to be easier to find out the potential at
that point due to all the different charges;
I can add those potentials because these are
scalar quantities and then you find out the
gradient of that that gives me the electric
field.
So these are the very basic qualities for
representing the electrostatic parameters:
the electric field and the electric potential.
When you meet next time then we will try to
now establish the basic laws which connect
the electric displacement and the charges
which are responsible for creating the electric
displacement. And then subsequently a similar
analysis we will carry out for the magnetic
fields for the magnetic flux densities.
So let us summarize what we did today. Today
we saw saw some of the more vector operations,
we saw two very important theorems for the
vector integration what is called divergence
and Stokes' theorems which converts the surface
integral to volume integral and line integral
to surface integrals and then we also saw
other basic relationships between the displacement
vector and the electric field and a parameter
what is called permittivity of the medium
and how it changes for isotropic medium to
anisotropic medium.
