The curl - ElectroMagnetic Field Theory Video Lecture - Electronics and Communication Engineering (ECE)

aha
they just feed all of us belarius
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some tea
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good morning last lecture was a very
heavy lecture that introduced lots of
new mathematical concepts so this
lecture I am basically going to re re
work through those concepts make them a
little more sensible hopefully so that
you feel comfortable with all the things
I did so let me put down what what we
did we talked about the magnetic field
and we say that the magnetic field which
is earlier if God is mu naught over 4pi
volume integral J cross R 1 2 over R 1 2
cubed VB prime
we took this expression and we
identified that this piece R 1 2 over R
1 2 Q is nothing but minus of the
gradient of 1 over R 1 2
this came from electrostatics because
this was found in electric field this
was found in potential so if - or the
gradient of potential is electric field
and the only difference between the two
expressions is this then it must be true
that is all coming out of the fact that
- of the gradient of 1 over distance is
distance over cube of its amplitude so
it is a it is actually mathematically
this is the most important relationship
that there as everything else in
electromagnetics so you take this you
put it in there and you come up with a
new result you come up with a result
that B is equal to a new operator
that B is the curl of something else and
what exactly is this curl this curl is
nothing but you take the vector product
cross product of two vectors and you
replace one of these vectors the first
one by the operator gradient so for
example instead of saying V cross W
which would be X hat V Y W Z minus VZ w
y plus other terms now we replace it as
curl of W is X hat this V Y becomes del
Del Y of W Z minus this V Z becomes del
Del Z of W y plus dot dot dot so we are
treating gradient the gradient operator
as a vector and when you do that this is
now only a function of one variable one
vector field the other vector vector
field has been replaced by a
differential operator so whereas this
was a function of two vector fields is
now only a function of one vector field
and it gives you back a vector field so
it is a vector operator it takes vectors
and gives you vectors and this operator
is called the curve
and we what we have seen is that the
magnetic field is the curl of something
of a vector field
so obviously curl is a very important
concept now we also saw is the last hand
that if you took this curl concept and
you looked at it closely what you could
say is supposing I go to a very small
loop in say X Y this is at X this is at
X plus Delta X this is at Y is a y plus
Delta Y and if you worked out what was
loop integral V dot DL for any vector V
any vector field V what you found we
worked it out
was that it is equal to curl of V times
the area
so in other words a definition for curl
of V is one over area
sloop integral V dot slash where this
loop is a very tiny loop its size is
Delta X Delta Y I am NOT going to repeat
the derivation because we did that now
why this is interesting is we have
already seen this somewhere else we have
seen loop integrals in connection with
the electric field so in electrostatics
we had that e is equal to minus gradient
of Phi and the very same derivation that
allowed us to write this which was that
integral e dot DL from position 1 to
position 2 is a function of 1 & 2 only
that is it does not depend on how you
went from 1 to 2 it depended on where
you started and where you end it so if
you took 1 to 2 and you took 2 to 1 and
combined them together you could prove
that loop integral e dot DL was always
zero no matter what loop you took a dot
DL is always 0 which means if you took a
very tiny loop yo DL is also 0 or curl
of E is identically 0
so in electrostatics we have is - the
gradient of Phi curl of e is zero and
divergence of E is equal to Rho over
epsilon so this expression which I have
been putting several times can now we
put more mathematically it is saying the
same thing saying this is equivalent to
saying this because the definition of
curl is nothing but 1 over the area loop
integral of e dot DL so saying curl of e
is 0 and saying loop integral e dot DL
is 0 is to restate the same facts but
this is a differential operator this is
an integral and there are times when the
integral is more useful and there are
times when the differential equation is
more useful so we use both of these so
you can see that in electrostatics we
have got two equations you have got an
equation saying curl of e 0 and you got
an equation saying divergence of e is
Rho over epsilon and we also know
something else we know that there is a
unique solution to this problem that
given this pieces these pieces of
information given some boundary
conditions there is only one vector
field which gives you these answers
which means curl and divergence are
enough information to fix
all right this is a very crucial
statement we will come back and make it
more precise but it is worth looking at
it we found out that divergence e was
something we found out curl of E is
something we have also found out that E
is spin down which sort of means that if
you have these two pieces of information
you have the vector field and the
statement to that effect is theorem will
prove later all right so what I want to
do today is to take this idea of curl
and make it more more sensible so all
very well to write down curl but what
does it mean so I am going to take some
examples and I am going to try and work
out those examples and show you where
curl is coming from and of course I will
also derive curl for non Cartesian
conditions so let us take simplest
example I have a uniform vector field
let us say the electric field
electric field is uniform so I know that
if I drew a little loop the loop is
tangential to the field only on these
two edges and these two edges it is
normal so DL dot V or DL dot E is zero
so I know that loop integral e dot DL
let me call this side one this side too
and let us say that I am going through
this loop in this direction so it is
equal to e at location 1 times Delta L
minus e at location 2 times Delta F but
E is the same at both places it is a
uniform field so it is equal to 0
what does curl tell us well to write
down curl I would want to actually give
a form for this so I will say that the
vector field is along the X direction
and it has some value e naught so let me
take the curl of that curl of e would be
equal to I will do the determinant X hat
Y hat Z hat del del X del del Y del Del
Z II X is e naught e Y is 0 ez to 0 so I
have to take the determinant of this
matrix and that should give me curl well
if I take the X component I have 0 and 0
in one row so that 0 X hat times 0 if I
take Y hat then I have del Del Z of Z
naught but this side is del Del X of 0
then I have a Z hat
del Del X of 0 minus del Z naught del 1
but II not is constant so it has no
derivatives and so this is becomes zero
so this is an obvious case we knew that
there was going to be no curl and we
found that our formula works out alright
now let us try to do a slightly more
complicated case it is again a case
where we know the answer I am going to
take the field of a point charge
now if you take the field of a point
charge what do you expect curl to be I
am NOT going to work out yet in
spherical coordinates because we haven't
worked out what curl is in spherical
coordinates but I am going to argue for
you that curl should be 0 we already
know that in a certain sense because we
know that loop integral e dot DL is
always going to be 0 by electrostatics
but let us work out that loop integral
supposing I had a loop that did this so
the loop is pointing out of the board
that is its faces out of the board and I
am going to try and go around it in this
direction the counterclockwise direction
well the electric field is normal to
this and this so 2 2 edges 1 & 2 the
electric field is normal so it does not
give me any contribution to size 3 & 4
there is a contribution so what is the
contribution it is minus e are x dr
because the electric field is pointing
outwards the loop is going inwards so
there is a minus sign or I should say er
x minus d r+ here also er signs plus BX
it is equal to 0 so obviously if I take
a loop of this type this electric field
cancels this electric field the loop is
going in opposite directions but the
electric field is the same if I take an
take a loop that is on the surface of a
sphere
well the electric field comes out of the
sphere so it is 90 degrees to the loop
at every point so clearly is just zero
times Rd theta plus zero times R sine
theta D Phi so it is zero so that
particular look does not give me
anything now I can look at the loop that
corresponds to a vertical direction
rather than a horizontal direction but
because this is spherical symmetry if I
if I look in the vertical direction
rather than the horizontal direction it
is still the same picture that is if I
cut the the three-dimensional picture
vertically I will still find that I have
got a charge in the middle and I have
got electric field pointing in all
directions so no matter what I do I am
getting integral e dot DL a zero
as I said this is what I expect
electrostatic fields cannot give me e
dot DL nonzero but it is good to always
draw these pictures and make sure now
let us go for a case where there is
curve let us try and understand what
sort of field can give me curve first
let me again look at a case where it
does not give me I am going to go to
Cartesian this is X this is y and let us
say I have a field which is uniform in Y
and growing
what sort of feel can this be for
example supposing you have a mountain
and there is water flowing and you are
looking at this water velocity of the
river so here it's low here it is
feeding up here it is very fast so the
water is flowing faster and faster is
they occur well you can draw your dotted
loop there is nothing on this side there
is nothing on that side because they are
90 degrees what about these two sides
well in these two sides you can do an
inn you can do an integration it is done
there is a net velocity here the same
net velocity here so when you do the
integration over this side it is equal
in magnitude to the same integration
done on this side because it is the same
vector at both points but your loop is
going this way here and it is going this
way so whatever you gain from your a dot
DL on this edge you lose on your a dot
DL on this edge so loop integral V dot
DL is plus 0 which means curl of V equal
0
now from the various examples we have
seen so far it should be fairly obvious
what you require for curl what you
require is a variation in the field
which is at 90 degrees to the field
itself so that is the kind of example I
am going to draw next let's look at it
again X Y I am going to take a field
which is weak and it is getting stronger
and it will get stronger and stronger
as you go further
now what happens well let me draw my
loop as usual this side and this side
cannot contribute because the field is
90 degrees to DL so e dot DL goes to
zero but now I have sides one and side
two and let us say that I am doing my
integration this way so I have integral
over one eat or DL plus integral over to
e dot DL now clearly in the region 1 e
is larger right so it is DL times e in 1
here it is plus minus dl e at two but e
at one is not equal to e a to yet one is
a large vector e a two is a small vector
so these two are not equal this is
bigger than this so even though they are
tending to cancel the cancellation won't
happen you will get G L times T of 1
minus e of 2 and it is not equal to 0
this is a case where you have curve
there is a an exact aynd of an
experiment you can do which is mentioned
in every textbook what you can do is
imagine that you made a paper boat and
you put it into this vector field just
imagine it in your mind and you watch
what happens to that paper board or
better than a paper would put a flower
so you can see that the side of the
flower at the top is being dragged fast
the side of the flower here is being
dragged back slope is being dragged
forward slowly so what happens is one
side of the flower gets pulled further
the result the flower starts rotating
and the rotation of the flower is
nothing but the presence of curl now in
order to fix this in your mind I am
going to take an actual expression for
this field I am going to say that I have
say an electric field or Y an electric
field I will choose a magnetic field
which is equal to in the X direction but
its magnitude is y so if 0 at y equals 0
and it keeps growing and gets bigger and
bigger so that is very similar to this
view now I want to take the curl of this
field so what do I get if I take the
curl
it is the determinant of X hat Y hat Z
hat del del X del del y del del Z the X
component of B is y the y component of V
is 0 Z component of B is 0 so this is
the determinant I want to calculate let
us do it X hat is the determinant of
this 2x2 piece is 0 Y hat the
determinant of this 2x2 piece is del Y
del Z minus 0 and the Z component is the
determinant of this piece which is 0
minus del Y del Y well del Y del Z is 0
because if I move in the Z direction the
y coordinate does not change that is the
meaning of coordinate systems if I move
along one coordinate the other
coordinate is not changing so this is 0
but del Y del Y is equal to 1 if I move
unit distance along Y the value of y
changes by unity so this slope of Y with
respect to Y is 1 so it is equal to
minus Z hat so curl of B is not 0 and in
fact curl of B is minus Z hat does that
make sense well you can see take a
screwdriver in your right hand and try
to screw in the direction that you think
the rotation is going to happen imagine
your flower put it into this field you
know that the flower is going to rotate
this way it's not going to rotate the
it's going to rotate clockwise so if you
take a screwdriver rotate it clockwise
it will go into the board but now what
is Z hat Z hat is X hat cross Y that is
the unit vector along Z is a unit vector
along X vector product unit vector along
Y so I take my screwdriver again but I
now need to turn it from the direction X
to the direction Y can I do it this way
no I have to do it this way because I
have to bring it from X to Y so the z
direction is coming out of the board
curl is going into the board and that is
why we have the minus sign so this is a
case of a vector field that has curl now
of course the common examples of curl
are not in Cartesian fields at all the
Communist example that you would have
seen is your wash basin you would have
you have water coming from a tap and the
water fills up and then you allow it to
drink so the what does the water start
doing water starts circulating and then
leg comes out of the drain comes through
the pipe and goes to the drain this is a
classic case of circulation
so it is these fields that were first
the reason for calling it curl because
you could see that the vector field was
curling round and round
there is no curl in a Cartesian field it
just so happens that the mathematical
operation gives you curl but when people
first invented the operator they were
thinking of Eddie's they were thinking
of whirlpools
so let us look at a whirlpool kind of
field and see what kind of whirlpool
feel has curl and what doesn't so I am
going to draw now in our theta Z it is
out of the board so this is my X Y so
some general direction is R and theta is
is this direction now if I want to draw
a circulating field I am going to draw
arrows like this
now the question is what kind of feel of
this type would have curl and what
wouldn't so I am going to guess well
what about if I look at this I will say
what about a vector field V which is
equal to theta hat
so it is
it is not growing in amplitude but it is
still always rotational so for this we
have to draw our a dot DL of e dot DL so
we make a loop I have made a loop where
the sides are 45 degrees and 0 going
from R to R plus Delta R well you can
see that the pieces at constant R are
the ones that are parallel to the field
pieces that are along constant theta on
90 degrees to this field so these two
sides will not contribute exactly
opposite of electrostatics in
electrostatics the field was radial now
we have got fields that are in theta but
what about these two fields this
distance is R Delta Theta where Delta
Theta happens to be 45 degrees in this
diagram this distance is R plus Delta R
Delta Theta and now I am going to do I
have to choose a direction in which I
will go around so let us say I chose
that direction so what will my loop
integral e dot DL be or sorry B dot DL B
there is the piece at R the magnitude of
V is 1 so it is 1 times R Delta Theta
and there is the other piece which is
plus 1 times minus R plus Delta R Delta
Theta the minus sign comes because theta
is in this direction but DL is in the
opposite direction so I get
delta theta times R minus R plus Delta R
R's will cancel so I get minus Delta R
Delta Theta
now to get curl I must divide by the
area of this object of this loop what is
the area area is equal to Delta R times
this length which is R Delta Theta so it
is all times Delta R Delta Theta and if
you look at the curl it has only a delta
R Delta Theta so curl of V is equal to
well I put a minus sign so curl of V in
the direction that is normal to this
which is into the board okay n is equal
to minus 1 over R times R Delta R Delta
Theta divided by R Delta R Delta Theta
what did I do this is the value I have I
multiplied and divided by R an area is
our Delta R Delta Theta so these two
cancel out so I get minus 1 over
so the amount of curl is minus one going
into the board well as it so happens if
you look at our hat and theta hat this
way z hat is coming out of the board so
if you wrote out what curl of V is it is
actually equal to plus one over R in the
Z direction so this is a field that does
have circulation if you put a flower in
here what is going to happen the flower
will as it moves will see this velocity
it will also see this velocity but is
moving much less here knowing much more
there as it moves is going to start
rotating and what you find is that this
has a much greater effect because it is
applied over a much longer distance this
is a much less effect and therefore the
flower starts rotating and as it rotates
it shows up that there is a curve
so we have seen that there is a
circulating field with curved other
circulating fields without curve I am
going to take a field which is large
near the origin gets smaller as you go
further away from the origin and get
smaller still so the particular field I
am going to look at is V is equal to 1
over R theta hat now what happens if I
do my loop integral the same thing that
I did earlier works except that I get if
I if I draw my loop I find this is let
us say 1 and this is 2 V of 1 times R
Delta Theta plus V theta of 2 times
minus R plus Delta R Delta Theta
that is just the definition the sign
comes from the fact that V is always
pointing this way but DL is pointing the
opposite way but V theta is now one over
R so it is equal to one over R times R
Delta Theta minus one over R plus Delta
R R plus Delta R Delta you can see the R
cancels out the R plus Delta R cancels
out so you get Delta Theta minus Delta
Theta equals zero so this is a case a
field that is getting weaker as you go
out specifically as one over R this is a
field that has zero curve so it is not
true that every circulating field has
curl it is not true that every straight
field does not have curl curl comes from
your imagination that you put in a round
object and does it start rotating if it
starts rotating it is curved curl is
present it does not rotate curl is
absent now you cannot keep on drawing
EDL's well actually you can but
mathematically you would like to
formalize and find out a mathematical
way of calculating what curl is your
appendix a in your textbook
give you the answer supposing you have
we have introduced this idea before but
I'll repeat it
supposing you have instead of X Y Z I am
going to call them u 1 u 2 u 3 and DL
along u 1 is H 1 u 1 Delta u 1 this will
be H 2 Delta u 2 and it is 3 Delta u 3
let me give examples for car TCM H 1
equals 1 h 2 equals 1 h 3 equals 1 u 1
is nothing but X u 2 is nothing but Y u
3 is nothing but Z for cylindrical u 1
will be R u 2 will be theta you three
will be Z if you look at distance along
R distance along R is nothing but D are
so H 1 is 1 distance along thither is
not d theta
it is Rd theta so H 2 is equal to R
distance along Z is DZ so H 3 equals 1
if you go to spherical your you one is
our u 2 is theta nu u 3 is Phi
distance in spherical coordinates along
r dr so H 1 is unity if you go along
theta
so that means this is my axis I go on a
great circle so if I go on a great
circle by a distance Sisters obtained it
by delta theta this distance is R Delta
Theta so H two is equal to R if I go in
the Phi direction then I am going on a
circle that is a fixed latitude and this
is the angle Delta Phi this distance is
Delta Phi times this radius which is R
sine theta so H 3 is equal to R sine
theta so very you can do it for any
coordinate system but these are the
three common coordinate systems so these
are the three things you should know now
in terms of this notation we have
already written out what gradient was
gradient of any I shouldn't use Phi
gradient of any scalar field is equal to
1 over H 1 del F del u 1 along the
direction u 1 plus 1 over H 2 del F del
u 2 along u 2 plus 1 over H 3 del F del
u 3 along u 3
in Cartesian is these three are missing
but in cylindrical for example you have
1 over R del F del theta so that is
where the 1 over R came from similarly
we have done it already for divergence
but now I want to talk about curl the
derivation is easy but I am NOT going to
give it it is more useful to know the
answer answer is if you draw we can
write down what curl is it can be
written as 1 over H 1 H 2 H 3
determinant of H 1 unit vector along u 1
H 2 unit vector along u 2 H 3 unit
vector of u 3 del del u 1 del del u 2
del del u 3 and H 1 V 1 H 2 V 2 H 3
so all these metric coefficients these
things that relate change in coordinate
to distance along the coordinate line
come in in this fashion one over H 1 H 2
H 3 is like a volume factor then your
unit vector now has a H associated with
it
because when you what what does it mean
to go one coin one unit in that
direction and your function itself is
also multiplied by H 1 H 2 H 3 let us
work it out for cylindrical coordinates
and spherical coordinates so what you
will get is for cylindrical one over H 1
H 2 H 3 is 1 over our determinant R hat
our theta hat Z hat del Del R del Del
theta del Del Z be our RV theta
VZ
that is what it is I just substituted h1
h2 h3 to get it so what does it become
if I take the R hat component it becomes
1 over or I'll say R hat over R del VZ
del theta minus R times del V theta del
Z then if I take the theta component the
R and the R cancels out
I get del V R del Z minus del VZ del R
and I'm going to write the last
component up here plus red hat divided
by R times
del Del R of RV theta minus del del
theta of
so this is just the expression there is
nothing magical about it you can in fact
derive it what is useful is it in fact
allows you not to keep thinking of e dot
DL I do not know if there is a good
thing or a bad thing because the e dot
DL the V dot DL is the correct
fundamental definition of curl so if you
can keep little loops in your mind you
can never go wrong and this expression
is nothing but little loop with its face
pointing along Z little loop with its
face pointing along R and little loop
with its face pointing along thither
each of these is nothing but loop
integral V dot dl divided by area that
is all it is so if you could directly do
it you do not need this expression but
you can take it as an problem and just
verify that the previous examples we
have done they all fit and give you the
correct answers that we have already
derived from loop integral for example
if you take 1 over r theta hat so what
happens this V theta is 1 over R so 1
over all term times R is 1 there is no V
R there is no VZ obviously if you have a
constant vector and you're taking
derivatives answer is going to be 0 so a
velocity vector with 1 over R theta hat
has 0 curl if on the other hand your V
theta was constant then this would have
been R because constant times R now if
you do the expression what you get is d
dr of r this term survives this goes
away this goes away and all the others
are 0 so you get one term which is d dr
of r and it is along the z direction
so what you get when you work out
this term survives and you get Z hat
over R times the day our derivative of R
which is one which is what we got
earlier we found that it's in the plus Z
hat direction and it's magnitude was 1
over R there is nothing surprising about
any of this because the loop integral
concept is behind this formula the
earlier derivation was more fundamental
than this this is only a formula if you
actually could put loop integrals and do
it that is the most correct way of doing
it all right ok I hope you are now a
little more comfortable with what curl
is now let us get back to LA
electromagnetics we had already seen
electrostatics told us curl of e is 0
and we had worked out that B is equal to
curl of a where a was equal to MU naught
over 4 pi volume integral J prime D B
prime divided by mod R minus a what I
call r12 and this is a at our
now we would like to know the properties
of B we say that B is equal to curl of a
so the first thing we would like to do
is to work out what the divergence and
curl of B are because we already know
that when we wrote down divergence of P
is equal to Rho over epsilon not somehow
we managed to fix e we got a unique
answer so we would like to know what the
divergence and curl of B is so when you
say when we take the divergence of B it
is actually the divergence of the curl
of some vector a and you put it into
your matrix and I am working in
Cartesian coordinates X hat Y hat Z hat
dot divergence so the X hat Y hat Z hat
gets replaced by del del X del Del Y del
del Z
this is what you get is the determinant
of a matrix with two identical rows now
those two rows are actually differential
operators so it does not really mean
anything to say they're two equal rows
but if you look at what will happen you
will see that if you choose to expand
along the third row then you will get ax
is acted upon by del squared del Y del Z
minus del square del Z del Y similarly
if you take a Y you will get you'll have
to check the sign I think is del squared
del Z del X minus del square del X del Z
CY plus for easy it is del squared del X
del Y minus del squared del Y del X
acting on Z and from your mathematics
you know that del square del Del Y del Z
is nothing but del square del Z del Y
these two are the same operator so this
goes off this laws of this cosine so it
is equal to zero
so the divergence of B is zero whenever
B is a curl and we have derived that B
is occur so we have got one as one of
the two relations you want you know what
the divergence of B is now that we have
one other fact which is that if you look
at how we derive for electrostatics that
loop integral e dot DL was zero actually
you can easily show that curl of the
gradient of anything is zero it comes
from the same idea that is the curl is
nothing but X hat Y hat Z hat del del X
del del y del del Z del sy del X del Y
del Y del C del Z so if I take say the X
component is equal to Del square Phi del
Y del Z minus del square sy del Z del y
plus e2 other terms and once again
because del Y del Z is the same as del Z
del Y this is 0
so there are two very important
statements that are there in vector
analysis and they are crucial to
electromagnetics one is divergence of
any curve zero and curl of any gradient
is zero this is why curl of the electric
field is zero this is why divergence of
the magnetic field is zero because b is
curl of a he is gradient of Phi now what
does this mean we already know that B is
the curl of something so supposing I
said B is curl of a and then I look at
this I can say B is also the curl of e
plus gradient or something because curl
of the gradient of anything is zero this
will just give me zero so the value of a
is not unique we have given an
expression for a but that expression is
not unique in fact there are many vector
fields a whose curl gives me B now if I
take the divergence of B I get zero
because it is curved I mean it is
already occur but if I take the curl of
B is going to be equal to the curl of a
curl of a plus
gratz I now I am NOT going to give you
the vector algebra right now but it
turns out that this is equal to gradient
of the divergence of a plus h I minus
del squared of e plus grad son and
because of this you can use grad SCI to
adjust the values that are present here
and by doing this we will be able to put
this a in the correct form where curl of
B will have a useful value divergence of
B is 0 which fixes B as as a unique
vector field it's a bad place to stop
but I will continue next time and
complete this topic
you