in the previous lecture we talked about a
integral form of ampere's law and used dell
cross b equals mu not j or its integral form
which was b dot d l is equal to mu not i enclosed
by that loop to calculate magnetic field in
certain symmetry situations we also said there
the question we are traced was can we 
define a potential for magnetic fields in
this lecture we address this question recall
that for electric fields we had curl of e
equal to zero and we use this to define a
potential because curl of gradient is always
equal zero one could write because of this
condition at e is equal to minus gradient
of v r because this curl will always give
you zero similarly now we use in the context
of magnetic field this equation that the divergence
of magnetic field is always zero no one can
show that divergence of curl of a vector is
always zero
so divergence of b is zero implies where i
can write b as curl of a of r and therefore
now we can say that b can be obtained as the
curl of another vector field a which i am
going to call the vector potential so in the
case of electric field we had a scalar potential
v r which is also interpreter as the work
done in the case of magnetic field it is a
vector potential whose curl gives me magnetic
field now just like the v r was defined within
a constant what is the mean that means i could
have v r or v r plus c i could had a constant
v r it's still a give me the same electric
field of same physical quantity in a similar
manner when i have magnetic field b r as curl
of a r i can also write this curl of a r plus
gradient of some other quantity let's chi
r because curl the gradient is always zero
and therefore a r is defined within within
variant of a scalar field
so just like the potential b had an arbitrary
of this constant a r has an arbitrary up to
the gradient of a scalar field what about
physical 
interpretation of a can i ask what does a
mean recall that the potential energy in the
case of electric field was the work required
to more a unit charge in that electric field
can i have a similar interpretation here in
the answer is not simple but we can get some
inside side while looking at dimension of
a 
curl of its give me b and therefore a as dimensions
of b times a length b you call the we have
q v b equals force and therefore b is force
over q v
let us substitute that here then i get force
times length divided by q v as a dimensions
of a and therefore q a as a dimension of f
l divided by v i can write this time f times
b is a moment of 
so q a as dimension of momentum although is
not directly moment of particle or anything
but we let us that this display a role right
now will leave it at that let us now take
examples and solve some certain problems where
we calculate the magnetic electro potential
example one i will take a straight current
carrying 
wire of infinite length in this case we just
saw in the previous lecture and we have calculated
earlier also that be filled looks like going
around in circles like this
and therefore i can write b is equal to b
r a distance s is equal to mu not i over two
pi as in the direction pi this is nothing
but curl of a since there is a cylindrical
symmetry i use the definition of curve in
terms of cylindrical coordinates and cylindrical
coordinate since i want only the five component
curl of a a phi component is given as partial
a as component z minus partial a z partial
s and this in our case happens to be equal
to mu not i over two pi by s this immediately
suggest that i can take a s to be zero and
a z is mu not i over two pi logs with the
minus sign in front you can immediately see
this gives me the right answer for b i can
also choose a phi to be zero and therefore
i can write in this case of an infinitely
long current carrying wire a vector to be
minus mu not i over two pi log s in this direction
example two let us take a uniform field b
equals b z so that like this every where i
am using cartesian coordinates i am going
to write this as x y z partial by partial
x partial y partial z a x a y a z i am interested
in only the z components so there will be
some x component that's why x component which
is partial a z partial y minus partial a y
partial z plus y component partial a x partial
z minus partial a z partial x plus z component
which i am really interested in partial a
y partial x minus partial a x partial and
this should be equal to this quantity so i
want partial a x over partial y partial a
y over partial x v equal to b you can immediately
see that here three different possibilities
i can take a y to be equal to b x by two and
a x to be equal to minus d y by two
so that a vector is equal to b by two y x
plus x i y direction which i can write as
b s by two phi with a minus sign this one
possibilities second possibility i can have
a y equals b x a x equal to zero sorry i show
you a z equal to zero here and a z is and
equal to zero so i have a equals b x y or
third possibilities a y equals to zero a x
equals minus b y and a z equal to zero so
that a is minus b x b y in the x direction
all this three possibility give me the same
answer for b as i said earlier they will all
differ by quantities which are gradients of
one over the other quantity let's now look
at two particular cases i had a equals minus
b y over two in the x direction plus b x by
two in the y direction
and i also had a equals minus b y x let's
take the difference this call this a one a
two a one minus a two b y by two x plus b
x by two y this you can clearly see as a gradient
of b x y by two so you shown that the difference
between the two vector potential is equal
to gradient of a scalar fieldin third example
let us calculate the vector potential for
this current carrying sheet with current surface
current in k equals k y let me remind you
for this current carrying sheet the previous
lecture we had x is in this direction y going
along the sheet and z going up and the magnetic
field b was given as k mu not k over two x
or mu not k over two x with the minus sign
right
so let's calculate a for this a gives me as
such that gives me field mu not k over x ok
mu not over two in the x direction for z greater
than zero x component is partial a z over
partial y minus partial a y over partial z
so i can immediately write there either a
z is equal to mu not k y over two and a y
is equal to zero or any other combination
that gives me thois answer if you calculate
the difference between the different a's again
that come out to be equal to gradient of certain
scalar field
