we have seen that the magnetic field is produced
by current and for study currents that means
this current which are not changing with time
is given as d l prime cross r minus r prime
over r minus r prime cube this also now tells
the us properties of magnetic field so this
is known as bio savart law and this for those
cases where d i d t is zero now remember when
we we will discussing electric field we said
that any vector field can be calculated by
is divergence and curl together give a vector
field so now if you want to work with magnetic
field its important to know what the divergence
of b s and what the curl of b s this lecture
is going to be concerned with these two quantities
i am going to use a little bit of vector calculus
and i will keep explaining it as we go along
that is necessary and i would also request
you to master that technique because make
life very easy with this fields so when i
have b r equals mu zero over four pi i d l
prime cross r minus r prime over r minus r
prime cube let us calculates its divergence
directly 
divergence of b is going to b mu zero i over
four pi divergence of this quantity d l prime
cross r minus r prime over r minus r prime
cubed notice this is divergence with respect
to r because this divergence with respect
to the r what is quantity integration respect
to the prime quantity so i as here which is
the which is where field is being determined
so this is respect to r of this quantity integration
is respect to the prime quantity so i can
actually take it inside and write this mu
zero i over four pi integration of divergence
of this vector product d l prime cross r minus
r prime over r minus r prime cubed
now from the vector identity the divergence
of a cross b is equal to b dot curl of a minus
a dot curl of b we find that divergence of
d l prime cross r minus r prime over r minus
r prime cubed is going to be equal to r minus
r prime over r minus r prime cubed dotted
with curl of real prime minus d l prime dot
curl of r minus r prime over r minus r prime
cubed notice that d n primers with respect
to very well r prime and this differentiations
is respect to r so this term goes to zero
for the second term recall that r minus r
prime over r minus r prime cube is nothing
but minus grad of one over r minus r prime
so this term is d l prime dot curl of grad
of one over r minus r prime what is curl of
grad curl of grad is again zero so this whole
term comes out to be zero
and therefore what we find is that divergence
of b r is nothing but zero this is a result
we have anticipated in one of the previous
lectures by looking at that there are no mono
poles there are no single sources of magnetic
field so we find from the bio savart formula
that divergence of b at any point is always
zero this also means there are no independent
sources or sings of b let us see how does
that happen in the independent source and
fields will be coming out of here and this
divergence or b proportional to delta r or
equivalently by divergence therom flux to
the surface to be known sero tell me the divergence
cannot be zero
next this calculate curl of b directly so
this is also going to be mu zero i over four
pi curl is respect to r integration of real
prime cross r minus r prime over.over r minus
r prime cubed to calculate to calculate curl
i am going to play little check if i take
this current loop or current carrying wire
assure is likely wide at any point suppose
i take this point where i am taking d l prime
and this cross section is a then i can write
i d l prime vector as j at that point where
j is the current per unit area times a times
d l prime vector d l prime is in the direction
of current so i am going to this vector over
to j and define by current density is a vector
quantity j r prime a d l prime
what is a d l prime a d l prime is the volume
of this smaller element which i am not showing
by the shaded purple line therefore i can
write this whole thing as j vector r prime
d v prime all right so therefore i can write
write b r which is mu not i over four pi integration
d l prime cross r minus r prime over r minus
r prime over prime cube as equal to mu over
four pi integration j r prime cross r minus
r prime over r minus r prime cube d v prime
obviously the integration over j is not zero
and that precisely what how we transfer form
that i d l from to j d v prime formula
now let's take a curl so curl of b r is going
to be mu zero four pi curl of again i take
curl inside because curl is respect to r the
and integration is respect to r prime so j
r prime cross r minus r prime over r minus
r prime cubed d v prime now i am going to
use vector identity which is curl of a cross
b is equal to b dot del operating on a minus
a dot del operating on b plus a divergence
of b minus b divergence of a what it means
it let me explain in next slide i have written
curl of a cross b as b dot del a minus a dot
del operating on b plus a divergence of b
minus b divergence of a you have familiar
with this term and this term or this term
this term means is b dot del minus b x d by
d x plus d y d by d y plus d z d by z and
this operates on this vector a given a vector
point
now let's substitute for a i have j r prime
for b i have r minus r prime over r minus
r prime cube therefore b dot del operating
on j r prime this differential with respect
to r j is with respect to r prime is gives
me zero del dot b is divergence of r minus
r prime over r minus r prime cubed and that's
recall you did time and again is four pi delta
r minus r prime again a physical way of looking
at this quantity r minus r prime over r r
minus r prime cube is nothing but represents
electric field due to a charged four pi epsilon
not in magnitude and therefore the charge
density is the divergence is a charge density
which is nothing but a delta function right
so that's gives you that very quickly
let us ah similarly del dot a which is divergence
of j r prime is also zero so when i take the
curl of the which is mu zero over four pi
integration curl of j r prime cross r minus
r prime over r minus r prime cubed d v prime
curl of j r prime j cross r minus r prime
over r minus r prime cubed is nothing but
equal to r minus r prime over r minus r prime
cubed dot del operating on j r prime minus
j r prime dot del operating on r minus r prime
over r minus r prime cubed plus j r prime
times divergence of r minus r prime over r
minus r prime cubed minus r minus r prime
over r minus r prime cubed divergence of j
r prime
and we have fond this term is zero this term
is zero and we left with minus j r prime dot
dell r minus r prime over r minus r prime
cubed plus four pi j r prime delta r minus
r prime this is this term 
and therefore i have curl of b at r is equal
to mu zero over four pi integration of minus
j r prime dot del r minus r prime over r minus
r prime cube plus mu zero integration j r
prime the four pi four pi cancels delta r
minus r prime d v prime d v prime which is
equal to mu zero over four pi minus outside
j r prime is a prime dot del operating on
r minus r prime r minus r prime over r minus
r prime cube b prime plus mu zero j r j r
is a current density as you already discussed
the direction as there that of the current
and current percent unit area
the only term is now evaluate is a this term
to do this again i will to a check so i am
interested in evaluating j r prime dot dell
r minus r prime over r minus r prime cube
d v prime when del operates on r minus r prime
it is equivalent to over del operating on
any function on of r minus r prime it is equivalent
to minus del prime operating on one over r
minus r prime correct so i can write this
this you can clear on many function as you
see that this is true so i can write this
as integral j this is r prime r prime dot
del prime acting on r minus r prime over r
minus r prime cube d v prime to the sign
to evaluate this i calculate this x y z component
separately that if i do x components is zero
then you can check the same thing for y and
z component so x component of this is going
to be the minus sign j r prime dot del prime
since this is a dot product already taken
so there is no x component in this all three
components come here i have x minus x prime
over r minus r prime cube d v prime let's
go to the next page so i am evaluating is
integration j r prime doted with the del prime
component i take only of the vector quantities
x minus x prime over r minus r prime cube
d v prime
now let us look at the divergence of this
quantity with respect to del prime j r prime
x minus x prime over r minus r prime cubed
this is equal to divergence of j r prime x
minus x prime over with divergence if i integrate
both sides with respect to d prime this is
what i get notice that this term it preciously
what i want now for steady currents del prime
dot j r prime which other row as is minus
d row r prime d t is zero otherwise it will
be minus d row for prime d t nut write now
is zero for steady current only is the current
time wearing this time does contribute and
we see later that this screen size of something
on the displacement current for the timing
bio savart law for steady current this term
is zero
another current that localized that means
for away from the system they go to zero this
divergence can be changed into a surface integral
and that surface integral will also go to
zero because far away j has j becomes zero
on that surface so what you are left with
this the last term this term here since left
hand side is zero this term is zero this term
is also zero so what i have shown you now
if i go back is that this term here is zero
and therefore for steady currents we find
that curl of b is mu zero j the current density
r so what we found in this lecture through
bio savart law that divergence of d is zero
and this physically means no mono pole and
curl of b is mu zero j of r compare this equations
with that for the electric field where i had
divergence of e as row r over epsilon zero
and curl of e zero so here curl is b is not
zero and divergence of b is zero i remember
this is two only for steady current this time
dependent current and then there will be a
negation term here which will deal with data
right now will focus only on magnitude statics
and therefore magnatic fields you to steady
currents
