so far in this course we have focused on static
situations so we've dealt with electrostatics
which concerned itself with fixed charges
and corresponding electric field and the potential
v r or we dealt with magnetostatics which
dealt with steady currents news that means
the current did not depend on time j r current
density the corresponding magnetic field and
the vector potentials and so on now we are
going to go into dynamics 
and what that means is we are going to change
things with time 
and see what is the effect of this so first
thing that comes in this is faradays law
what faraday first thought should happen and
then observe is that if i have a circuit and
i change the flux of magnetic field passing
through it so we change d by d t of the flux
passing through it then this induces an e
m f and the units so you know as such that
e m f is equal to d by d t the flux the magnitude
of e m f alright what it means is that if
i have a circuit then because of this e m
f the current will start flowing in this current
starts flowing due to change in flux what
about the direction of current 
and that is given by lenz's law which says
that the direction is such that it opposes
because of change and we'll put that mathematically
in a minute
now what is means is that if i take a circuit
and let the area of the circuit change or
the magnetic field through it change then
there'll be an e m f generated let me show
this to you through a demonstration i want
to acknowledge professor h c verma at i i
t kanpur who has developed these demonstrations
for school going kids in the first demonstration
i have this tube on which a lot of wire has
been wound i put this magnet inside now if
i shake the magnet as it comes near the wire
the flux is going to change because as the
magnet comes nearer the field becomes stronger
and as it goes away field becomes weaker again
the flux is going to change and that should
produce an e m f to show that an e m f is
produce that wire is connected to an l e d
out here and this l e d should glow and let
us see that
you see the red light coming so as the magnet
is moving around its changing the magnetic
field and the change in the magnetic field
changes the flux and that generates an e m
f so this is a first that you have seen that
if i throw a magnet through a coil this is
which is what faraday did that produces an
e m f in that the second thing is that the
direction of e m f is opposite such that it
opposes the change so here we have a coil
in which we have put this again developed
in professor h c vermas lab put these iron
spikes which make the magnetic field very
strong here as soon as you will see as you
switch on the switch so that the current starts
flowing the ring in this will try to go away
because the current generated or e m f generated
in the ring is going to be such that it's
going to oppose the change so i'll just switch
it on and you'll see that the ring jumps out
and that is the lenz's law let us now how
do this mathematically
so what we have seen is that we have e m f
which is equal to d phi d t and to show the
lenz's law i put a minus sign in front you
may ask how does this minus sign help and
lets discuss that a bit so e m f is equal
to minus d phi over d t which is equal to
minus integration b dot d s d by d t where
b is going through this area passing through
this area and d s is the area element in this
area so b dot d s integrated over gives me
the flux there are two ways that this can
change either b can change with time so i'll
put d by d t with the minus sign dot d s and
the area remains fixed or i can have b constant
in time but this area changes that means area
goes in and out and therefore the flux changes
this is due to movement of the boundaries
whatever e m f comes is sometimes referred
to as motional e m f and this is due to the
change in magnetic field
so first let us look at an example of motional
e m f you have very familiar with if i take
a wire and put a rod on this metallic rod
in which the field is coming out and i pull
this with velocity v then the area is changing
and this change in area gives me an e m f
which is equal to b v times l where l is the
length of this rod v l gives you the rate
of change of area and the direction of current
is going to be such that it changes it opposes
the change so the current should be such that
it opposes the rod end it opposes this being
pulled out so i v is coming out i cross b
it will give me a full force like this therefore
i should be going down in the rod and therefore
in the loop it should be going like this
another example of this would be i take a
battery for a switch and put it through a
coil so that when the current passes through
the coil there's a flux through it and any
change in current would change this flux now
e m f is going to be proportional to or equal
to d phi by d t with the minus sign now i'll
make this minus sign mathematically meaningful
phi you know is going to be equal to in such
a coil is going to be proportional to i and
this is proportionality constant is usually
called the not usually its always called the
self inductance so i am going to put an l
out here so this is minus l d i by d t now
we'll see the importance of this minus sign
so as soon as i put on the switch the equation
i am going to have is e m f which is e m f
equals v v the other e m f which is generated
in the system is minus l d i by d t is equal
to should be equal to r i if r is the resistance
of the system suppose v was zero and there
is an initial current i equals i zero in that
case this equation becomes zero minus l d
i by d t equals r i or l or d i by d t plus
r over l i equal to zero this would happen
for example if in the circuit there was a
current flowing and i suddenly took switch
off now this solution gives me i equals i
naught e raise to minus r over l t now notice
if instead of this minus sign here if i had
a plus sign if i had a plus sign i'll get
l d i by d t equals r i and this'll give me
an answer i equals i naught e raise to r over
l t in this case after i turned the switch
off it is the current is increasing with time
which is in violation of what we observe it
is in violation of energy conservation in
this case it is decreasing with time and eventually
going to zero so you see the importance of
that minus sign out here and that is the statement
of lenz's law
the second example through which i show the
significance of this minus sign is what we
have already solved is that this metallic
circuit on which i have this rod which is
being pulled to the right this is velocity
v and we have already taken that field was
coming out of this screen and in that case
we saw that the current was going clockwise
that is what opposes so this is what we physically
saw let's see it mathematically so i have
minus d by d t of b in this case it's a uniform
feel so i can write it like this b d a is
equal to e m f which in this case i can write
as r i and to understand the significance
of this minus sign i am going to write i in
a slightly different manner i am going to
write as r integration i is there u where
u is the unit vector in the direction of the
current so here it's going to be like this
here it's going to be like dot d l where d
l is the lined element along the path divided
by the total length this keeps it r i now
l direction and direction of are related by
the right hand convention so if i take my
fingers around l clockwise or counterclockwise
thumb gives me the direction of a
now in this case b is fixed a is changing
so i am going to write this as minus b dot
d a by d t is equal to r i over the length
lets write this as l don't confuse it with
the self inductance that we saw earlier integrated
over the curve u dot d l let us assume i take
a also to be coming out suppose i took area
a to be coming out if i took area a to be
coming out u and l will be counterclockwise
like this as v is being pulled out as v is
being pulled out a area is increasing a area
is increasing d a by d t is positive and minus
d a by d t is pointing into the screen and
therefore this entire product minus b d a
by d t is negative because of this minus sign
here
if the current direction was also counter
clockwise just look at the right hand side
here if i was counterclockwise then u would
be pointing opposite to l o sorry u would
be pointing in the same direction as l u will
be pointing in the same direction as d l and
r h s will be positive you see the inconsistency
of the left hand side and the right hand side
on the other hand if u is pointing clockwise
if the current is clockwise then u and d l
are in the opposite direction and you get
the right sign so you see the significance
of this sign this actually relates to energy
conservation again because if the current
went [clock/counterclockwise]-counterclockwise
rod will keep on moving faster and faster
and faster because it'll be pushed out on
the other hand when it goes clockwise it is
stopped
so what we have covered in this lecture is
that change in flux by faradays law 
gives e m f which is given as minus d phi
by d t and this minus sign is given there
to show lenz's law its mathematical effect
is that it makes the current flow in such
a manner that it opposes the change causing
it
in the next lecture we will be turning this
whole faradays law into in terms of fields
