One of the jobs we are going to do is how
to calculate electric field. One is obviously
going to say that given a charge distribution,
I am just going to do this E at r is going
to be 1 over 4 pi Epsilon 0 integration dv
rho r prime over r minus r prime r minus r
prime cubed here.
Is this how I am always going to calculate
it, suppose I ask you different question.
Suppose, I go to a region of space here, where
I do not know about this charge, but I know
field at these points, let me make it fill
different color, suppose I know field at this
point. So, what I will say is, field given
at a boundary I do not know about the charge
distribution but I do know field about a boundary.
Can I calculate field out here in the rest
of the space?
Let us look at an example, so where you do
not specify the charge, how much charge the
spherical body carries. But, what I give you
is I hide that spherical body, I give you
that on this surface the field is all the
same on this spherical surface and it is magnitude
is given by some constant. Can I calculate
the field in the rest of the space? In this
case, it is very easy, because you going to
take the center of the red sphere, you are
going to say that E at the surface is; obviously,
1 over 4 pi Epsilon 0, because all is spherical
some charge inside divided by R square that
is the magnitude.
And therefore, E further out is going to be
1 over 4 pi Epsilon 0 Q over r square which
I can write as 1 over 4 pi Epsilon 0 Q over
R square times R square over r square. I have
multiplied and divided by capital R square,
which I am going to say E R times R square
over r square.
So, here I could argue, because of the nature
of the field, because all spherically symmetric
about a certain point. So, I could sense that
this is due to a point charge, but what about
if I am given the same surface, but field
is you know has arbitrary direction, arbitrary
magnitude on the surface. What would I do
then?
Then; obviously, what I need to do is, take
the field out here and using some sort of
a differential equation, find out what it
is next point, what it is at next point and
then keep integrating. So, starting from a
boundary, one can find field everywhere else
if differentials of the field are known. Now,
I already said why do we want to do that is,
that not necessarily every time I will be
knowing what the charge is somewhere. I may
be given a boundary and where I know the field.
How do I build it up from here?
So, if I know the differentials I can do that,
in other words what I am saying is:Does E
satisfy a 
differential equation that can be solved to
find the field starting from a boundary where
it is known? Is the point clear? So, what
we are doing is we are trying to find a differential
equation, so that if I know field on a certain
boundary, in a certain region I can build
it up from there. Now, let us see how many
definitions are there. E has 3 components,
because it is a vector quantity, we have E
x in x direction is y component and E z and
these are all functions of x, y and z, these
are all functions of x, y and z, x, y and
z.
So, if I am going from point to point, this
point to this point, this point this point
all three components E x changes as I change
x, but keep y and z fixed. That means, I am
taking a partial derivative with respect to
x, E x also changes if I change y or E x also
changes, if I change z and so do all the other
components. So, if I am at this point let
us say this is x, this is y, this is z if
I go from here in the x z plane, the E x component
may changes as I move here.
If I go in the y direction E x component may
again change, so that is described by these
three differential partial derivatives. Similarly,
y component will change 
with all the three coordinates and so will
the z component. So, in principle I need all
these three 9 differentials, three for each
component in order to calculate field at some
other point starting from it is value from
a given point 9 components 
and I should be listing all of them. But,
we are fortunate there is a theorem and I
am going to say without proving it the theorem
called Helmholtz's theorem.
And what it 
states is given a boundary, suppose I know
the perpendicular component off a field any
vector field at the boundary. So, perpendicular
component all the field is known, then I am
going to define and explain those quantities
later, if we know the divergence I will explain
its meaning in a minute which is the combination
of derivatives in this from. The divergence
that is sum of the x component derivatives
with respect to x plus the y component derivative
with respect to y and z component z derivatives
with respect to z.
If we know the divergence and another quantity
called curl, if we know these two quantities,
let me write its components. For example,
the x component of curl would be partial E
z by partial y minus partial E y by partial
z, y component will be partial z of x minus
partial E z by partial x and the z component
is going to be partial E y with respect to
E x minus partial E x with respect to y. If
I know this quantity the curl and if I know
the divergence I can calculate field everywhere.
So, let us this is Helmholtz's theorem and
if I know the perpendicular component on the
boundary, then I need to know only curl and
the divergence. Let us understand the meaning
of these and that is what the rest of the
lecture is going to be about today I am going
to focus on divergence of a vector field.
You heard the word divergent, divergence means
we say views are diverging, there is diverging
light rays coming. So, what basically it means
is particularly, because you may have seen
it in your 12th grade book, if light rays
are going away from each other, we say light
rays are diverging, if people have differing
views we will say they have divergent views.
But, for electric field we are not going to
be interested in this, what we are interested
is this diverging mathematical quantities,
light rays are diverging I could very well
placed light rays by vector field and say
these are the lines representing vectors of
this field we are considering. What we want
to get a view of what a feeling for what this
divergence means and define it and in a very,
very precise manner. When we say something
as diverging an example to define it would
be a fluid flow which I talked about earlier.
So, let us say this water or a fluid is flowing
and all over the place it has a different
current at different points and looks like
light rays diverging that this velocity also
diverging can we understand what this divergence
means, can I give it definite meaning. So,
let us see if I take a volume small volume
here, if whatever that fluid is coming in
whatever is leaving is equal. So, suppose
fluid coming in is equal to fluid going out
in that case I would say it is not really
diverging whatever comes in goes out.
On the other hand if fluid going out is greater
than fluid coming in I will say this divergent,
the fluid flow, the velocities of the current
divergent. Because, they take away much more
than they are bringing in or the other hand,
if fluid going out is less then fluid coming
in I will say this convergent. Because, that
is like negative divergence, let us see make
sense.
Let see light rays from a lens, because I
say are diverging and if I take small volume
here, in that case whatever light comes in
is also going out, it maybe may not be diverging.
But, on the other hand, if I take a source
of like a point source of light and light
is going out from here all around. If I take
a small volume around it and see the light
going out of this volume all over the light
is going out, so this is really divergent.
On the other hand, if I take a point here
we are all the light is coming in, then I
will say this is convergent as converging
to this point is diverging from this point,
here at this point there is really no convergence
and divergence they could be as we just discussed
divergence of this point, because everything
is coming out of here. So, this kind of gives
you a feel for what diversion behaviour is,
divergence behaviour is going to be when something
everything is going out or there is a net
outflow or there is a net inflow this negative
divergence or something going in.
