we've learnt about electromagnetic fields
and learnt that they are governed by maxwell's
equations and let me write them because i'll
keep using them they are divergence of e is
rho over epsilon zero curl of e is minus change
in the magnetic field this is the first one
is the gauss's law second one is the faraday's
law third is curl of b is equal to mu zero
j plus mu zero times the displacement current
that we have already talked about partial
e partial t and divergence of b is zero these
are sufficient to describe any electromagnetic
phenomena i must point out right now that
i am talking about these fields in free space
what we want to use these now for is to show
that number one the electromagnetic field
that means electric and magnetic field together
transport energy so if i have magnetic and
electric field together they transport energy
from one point to the other two electromagnetic
field also carry momentum and if they carry
linear momentum then r cross p is angular
momentum and therefore electromagnetic fields
also carry angular momentum
in this lecture i want to focus on the energy
so for this let us consider an enclosed volume
v with surface s i'll need both these so i
am specifying these on the surface there is
this unit vector n at each point and inside
this is charge density rho and there are currents
let an electromagnetic field exist around
it so that we have e field and b field in
this region what i want to consider now i'll
make this volume little thicker so that the
walls little thicker so that you see it clearly
what i want to know now is how much energy
is being pumptined by these fields into this
volume or how much energy is being taken away
the only way the energy can go in and come
out is through interaction of fields with
charges and currents in this we must keep
in mind that number one magnetic field does
no work on charges because the force due to
magnetic field is v cross b times the charge
q and the power which is v dot f therefore
is zero
so the only work that is done on these charges
and currents is by the electric field how
much work does the electric field do work
done by electric field is e dot j per unit
volume that is easy to see per unit volume
if i have certain charge and density is rho
r then the force on this due to the electric
field is going to be e times rho r and the
power delivered is going to be v dot f which
is v dot e times rho r and rho v is nothing
but j so power or the work done per unit time
is going to be j dot e
another way to look at it is if i have a wire
and if i take a very small element in this
then the power in this is going to be v times
i this is joules heating joule heating and
this v is nothing but e times this small length
l i is nothing but j times a and you can write
this as e j delta l a this is nothing but
that small volume v e j is e dot j because
the only thing that comes into delivering
power is the component of e in the direction
of j so two ways we have seen that the power
delivered is j dot e and therefore if i see
if i go back to my earlier volume in which
there is this charge density rho r and current
density j actually i am writing them separately
but actually rho r flowing is j then the rate
of change in the energy of this system inside
this volume is nothing but d by d t of the
energy content and i should remove this d
by d t this is nothing but e dot j integrated
over the volume of this region
let us now use the maxwell's equation which
says that curl of b is mu naught j plus mu
naught epsilon naught d e d t this gives j
is equal to one over mu zero curl of b minus
epsilon zero d e d t and therefore i can write
d w d t is equal to integral over this volume
d v e dotted with one over mu zero curl of
b minus epsilon zero d e d t which can be
written as d v e dot curl of b over mu zero
minus d v integral over the volume epsilon
zero e dotted with d e d t this term together
is nothing but one half e square d by d t
and therefore i can write d w d t is equal
to one over mu zero integral over the volume
e dot curl of b minus one half epsilon zero
e square d v recall that this is the energy
stored in the electric field in this volume
now i am going to use a vector identity which
says that divergence of e cross b is nothing
but b dot curl of e minus e dot curl of b
therefore i can write e dot curl of b as 
b dotted with curl of e minus divergence of
e cross b recall that curl of e is minus d
b d t and therefore i can further write this
as b dotted with d b d t with a minus sign
minus divergence of e cross b which can be
further written as minus one half d by d t
of b square minus divergence of e cross b
and therefore i come back to this equation
and write this as d w d t is equal to minus
one half integral b square over mu zero to
make it look better let me remove this half
and put two mu zero inside this is minus sign
d v minus again one half epsilon zero e square
d v so i have taken care of those two terms
that gave me magnetic and electrostatic energy
and then i have finally minus integral d v
one over mu zero e cross b divergence of so
i have gotten these three terms let us simplify
them
so again i am talking about this volume inside
which there are these currents and charge
densities and what we've gotten is that the
change of the mechanical energy inside is
equal to minus one half mu zero integration
b square d v minus one half epsilon zero e
square d v minus integral volume integral
of one over mu zero divergence of e cross
b i can now use divergence theorem and write
this term as i'll remove this and write this
as minus integral the surface integral of
one over mu zero e cross b where the vector
sign for this surface integral the n is coming
out this is the surface element pointing out
this way the inside this volume inside this
volume whatever change is taking place is
taking place due to these two these terms
one is the change in the electro the the the
energy in the magnetic field and electric
field inside and this second term to interpret
this let us write it slightly differently
i'll write d w d t that is the power going
power change inside the system plus d by d
t of total energy electro magnetic and what
we mean by this is the sum of the energy which
is here due to magnetic field and due to electric
field and i brought it to the left hand side
therefore its plus is equal to integral of
one over mu zero e cross b dot let me write
this as d s prime so that d s prime is nothing
but the just opposite of d s that means this
d s prime is going into the volume what does
this mean this means that the change of the
energy inside this volume which is equal to
the change in the mechanical energy plus the
electromagnetic energy is equal to something
going into the volume and therefore i'll call
this one over mu zero e cross b as the energy
flowing per unit area per unit time
notice that if e cross b is pointing into
the volume energy will be going in the left
hand side should be positive on the other
hand if e cross b is pointing outs away from
the volume this e cross b dot d s prime will
be negative energy will be going out and therefore
the energy the left hand side term will be
equal to something negative the energy inside
will be going down so this is the interpretation
of this term one over mu zero e cross b that
is usually written as s and known as pointing
vector 
and this is equal to energy flow per unit
area per unit time notice that for s to be
non zero both e and b have to be non zero
so it is only a combination of electromagnetic
field that means if there's an electric field
as well as a magnetic field then only this
energy flows otherwise it's not there
next we will do some examples to illustrate
that e cross b over mu zero is really the
energy flow and it satisfies the equation
d w over d t plus d by d t of e electromagnetic
equals integral d s prime dot s the pointing
vector
