ok this is a course on electromagnetic theory
and any course starts with coulomb's law so
we start with in this lecture we going to
start with coulomb's law
law for forces between two charges
then we are going to see how this can be confirm
that this is really true experimental verification
third then we are going to solve some examples
for forces between two charges
and finally in this lecture we going to talk
about the force between a point charge and
a charge distribution
this is the program for this ah lecture and
based on what we cover today i am also going
to give you an assignment where you solve
three or four problems employing this concept
so what coloumbs aw says is that if there
are two charges points let's take the point
charges right now
so there is a charge q one and another charge
q two separated by a distance r and for the
future because i'm going to develop notation
also let us write this is r one two that indicates
the distance between charge one and charge
two then the force between them the magnitude
force of course you know from your mechanics
course that force is a vector quantity the
force between them is magnitude is going to
be equal to and this depends on what units
we are going to choose for q one and q two
finally it is the s i unit that we work in
so i am going to write this in terms of this
s i units it's going to be four pi epsilon
zero where epsilon zero is known as the permittivity
of vacuum q one two over r one two square
that means if the distance between two charges
doubled because of this square out here because
this square out here the force going to become
by a factor of force what i just said forces
not only ah quantity which is as magnitude
and it also have direction what about the
direction of the force
now you learnt in your previous classes that
if q one and q two are of the same sign then
the force is repulsive
what it means is that the force is going to
be along the same lines as the line joining
the charges and they go to repel each other
and the magnitude being the same on the other
hand if they are [vocalized-] opposite sign
so let me write opposite out here if they
are opposite sign then force is going to be
attractive
what that means is in that case the force
is going to be in this direction towards each
other right
so i hope this point is clear but let me make
it clear by drawing this picture again i am
drawing two charges q one and q two separated
by a distance r one two and if the charges
are the same so for same i am going to use
the colour orange with the charges of the
same then the forces repulsive
on the other hand if the charges are opposite
sign when the force is going to be attractive
how can we write all this in vector notation
this is very simple way of writing it i am
going to write first
now write the whole thing force effector with
the magnitude one over four pi epsilon zero
q one q two over r one two square now if i
am writing it in the vector notation i have
to denote force on one or force on two let
us write the force on one which i denote here
by this subscript one here it is going to
be the unit vector r one two direction this
this hat this hat here denotes the unit vector
now you see that just write r one two vector
from going from one to two so this is r from
one to two the unit vector is going to be
in r one two direction of magnitude unity
if i write like this then you notice for someone
is in the direction opposite of r one two
so i have to put in minus sign out of here
which i can equally well write as one over
four epsilon zero q one q two over r one two
square r two one
where are two one unit vector form two two
one r two one should be unit vector you notice
that this takes care of everything signs direction
and everything in one shot if the forces are
the charges of the same then the q one and
q two this out here is going to be positive
and therefore this sign is the right direction
on the other hand if q one q four opposite
the sign changes so let's repeat this i am
going to take two charges q one and q two
and write everything with the vector notation
so let me write vector form one two as r one
two vector from two to one as r one two one
i follow this convention all throughout then
the force on one with proper direction can
be written as one over four epsilon zero q
one q two over r one to square r one two square
is the same as r two one square times r one
to two unit vector
and similarly force on two can be written
as one over four pi epsilon zero q one q two
over r one to square which is same as a r
two one square are from one to two unit vector
this gives repulsion attraction everything
because now we are taking care of the direction
so this is the coulomb's law it give the magnitude
as well as the direction of the force how
do we check the experimentally with coulomb's
law do experiment with point charges are there
point charges available with which you could
do experiment the answer is no you recall
that there is a similar law one goes of and
that is gravitational law
and what does that say that says at the force
between two masses f let's say the masses
are m one and m two and this is r one two
vector then f on one is going to be equal
to minus g e m one m two over r one to square
r from two to one unit vector
this is always repulsive in gravitation we
have mass doesn't have a sign but the form
of the law is same the way when could check
this law by taking too large masses and find
in the force of attraction [vocalized-] so
the force is in this directions this is always
attractive what one could do is that one could
take this masses solid masses and measure
the force between them we will see later that
this force can be written as if the two masses
concentrated at the centre so the force between
two perfectly spiritual masses the magnitude
will be equal to g m one m two over r one
two square where r one two is the distance
between their centre and of course the directions
is attractive so this one ah right can we
do a similar things to check coulombs law
let us see that so if i want to look at coulomb's
law i take too large masses and the easiest
to charge r metallic spheres and put charge
is on it let us say i put positive charge
here since they are sphere in isolation discharge
you are going to be all uniform distributed
if this two sphere is far away necessarily
charge on them is movable because some of
the charge them when you charge them charged
moved on to them however since this charges
are are mobile then the charges are closed
whether this is not going to the situation
this charge is going and they're going to
have more and more charges because they repel
further away then there are nearer
so here charges repel so lot of them will
go to back side of the charges sphere and
there will be a less charge in front with
the final picture that if i take this two
charges charge sphere that the charges are
going to be more concentrated here and less
out here more concentrated on the back side
and less in front as a result i cannot really
think of them being separated by distance
which is the distance between the centers
of this charge and i do not know what the
distribution is if i do not know what this
distribution is how do i figure out how far
are the charges so coulomb's law cannot be
checked simply by bringing two charge spheres
of closed to each other and then measuring
force between them a way out is very interested
the way out of this is at what you can show
is that if of course is one over r square
dependent which we are assuming coulomb's
law is then on a sphere then if i take a sphere
say metallic sphere and put charges on it
all the charges will come to the surface
with the result that there will be no charge
inside charge a charge metal sphere so insisted
again if a force is one over r square dependence
or if this particular case of the electric
force between charges is one r square dependent
all the charges that we put on a metallic
sphere will come to the surface so the experiment
that is done to check this is precisely this
you take a sphere charge it and then take
a sphere outside and connect by wire if the
charge on this is sphere is zero i know the
force field is one over r square and that
is how it is done and this is known as cavendish
cavendish experiment and this is confirm that
it is one over r square is the degree that
one over r square plus minus ten raise to
minus seventeen
so we know very well that this coulomb's law
is really one over r square so let's see what
we established one force between two charges
one over four pi epsilon zero q one q two
over r square where r is the distance between
them number two experimentally
verified because of one over r square nature
of force extra charge on a metal sphere with
all come to the surface