Force on Current Carrying Conductor Video Lecture - Physics Class 12 -

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now we will talk about the magnetic
force on a current carrying conductor
before sometimes we talked about the
magnetic force on a charged particle now
let us suppose that instead of a charged
particle if there is a current carrying
conductor that means there is a
conductor which is carrying some current
and if this conductor comes inside the
magnetic field then what will be the
force experienced by that conductor
right so let us we will derive an
expression for the force experienced by
this current carrying conductor XI till
now we saw that the magnetic force on a
single charge is given by Q into V cross
V right now here we have to assume that
instead of this Chuck single charge
there is a conductor which is carrying
some current let us suppose that we have
a conductor which is in the form of a
rod which is of length ay L this is the
length of the conductor and the cross
sectional area of the conductor is a we
assumed that there a let us assume that
some current is flowing through this
conductor let us say some current is
flowing through the conductor some
current I is flowing through the
conductor now current is flowing means
there are some charges which is carrying
the current so to make to keep things
simple as of now we assume that there
are only one type of carriers right now
there are one type of carrier say
electrons that means electrons are the
charge carriers electrons are moving
that is why current is flowing through
the conductor that is what we have
assumed right now but in reality there
are positive charges also which are
moving inside the conductor right so for
now we assumed only electrons and let us
say diadem number density of the
electrons is n number density that is
number of electrons per unit
is n number density means number of
electrons per unit volume right so that
means that what will be the total number
of electrons in this rod so total number
of electrons in this rod will be equal
to the number density multiplied by the
volume of the rod right because number
density is number of electrons per unit
volume so number of electrons will be
equal to number density into volume and
what is the volume of the rod volume is
nothing but length into area so this
will be n a a so this will be the total
number of electrons inside this rod so
that means due to the movement of these
many electrons current is flowing
through this rod like that means these
many electrons there are total these
many electrons and these many electrons
are moving inside this rod because of
which there is current flowing through
the rod now we have seen before that the
magnetic force on one charge is this
much so now you have so many charges
these many charges are there so what
will be the magnetic force on so many
charges right let us assume that drift
velocity of each electron is let us say
that each electron is moving with a
drift velocity VD we have talked about
drift velocity right in our previous
lessons when we were talking about
electricity so let us suppose each
electron has a drift velocity reading
therefore velocity of all electrons will
be how much it will be total number of
electrons into VD so this will be the
total velocity of all the electrons now
we know total velocity of all the
electrons therefore we can calculate
that total magnetic force on all
electrons what will be total magnetic
force it will be the total charge on the
electrons into the total velocity of the
electrons cross the magnetic field so
what is the total charge on the
electrons the charge on the electron
equal to Q let us assume that the charge
on the Left I am I am now writing it as
e instead I am writing the charges Q and
what will be the velocity of so many
electrons it is nothing but in a L
reading like so the total charge on so
many lectrons will be n in L into Q and
this is VD cross B right so this will be
the total force on all the electrons
right so what I did basically we know
that magnetic force on a single charge
is Q into V cross B but in this case you
don't have a single charge you have so
many electrons inside this so we first
calculated how many electrons we have so
we have these many electrons now what is
the velocity of so many electrons so the
velocity of so many electrons is this
much so the force will be equal to Q
into NK LVD so instead of V it will
become n a n VD cross V so that will be
the total force on so many electrons
right so let me write it more neatly so
this is equal to Q n a l VD cross B
right so this is the total force on all
the electrons now from our previous
studies we know that current density J
is equal to n q VD right this is how we
define current density you remember in
our previous class C I cannot refer to
everything I cannot prove everything now
because I have already discussed all
these things in our previous classes so
I can replace NQ VD with this current
density J so what am I left with
I am left with FM is equal to j al cross
B now again here what is J current
density is current per unit area so
current density is current per unit area
so that means current density into area
nothing but current so this I can write
it as current and this is L cross B so
your magnetic force is equal to I L
cross B that is current into the cross
product of length and magnetic field so
here in this case your magnetic force
magnitude of magnetic force will be
nothing but F M is equal to I L cross B
please make a note that here length is
the vector it is the length vector and
because current is a scalar quantity
right and length is at the length vector
like how we talked about area vector
similarly here it is length vector
because if the orientation of the length
changes then the force experienced when
burning then so here also so the
magnitude in this case will again be
given by AI n B sine theta and when I
talk about dead where theta is the angle
between l and B that is the angle at
which this kind conductor is oriented
with respect to the magnetic field and
when I talk about direction the
direction is again given by L cross B
that means use right-hand rule you turn
all your four fingers from L towards B
then your thumb will give the direction
of the magnetic force right so I hope it
is clear to you now and another thing I
would like to mention here is sometimes
students get confused that here it is
written Q into V cross B but here
suddenly you this bracket is removed
why is it so that's because when you
look at your cross product let us
suppose you have some scalar quantity or
some number to say 2 into a cross B what
does it mean it means 2 into a V sine
theta right this is just for your
reference now instead of this if you
have to a cross B what does it mean it
means to a into B sine theta right so
the same thing applies here also here it
was Q into V cross B
so this instead of this V this Q into V
cross B rewrote it as QV cross B so QV
we wrote it like this cross B clear
so instead of QV I wrote this entire
thing as Q into V cross D so that is how
I wrote it here also so your force
magnetic force on a current carrying
conductor becomes I into L cross me
please remember these conclusions which
we are getting because when when you
reach towards the end of the lesson you
will see that you will have to find out
the direction of magnetic force on a
current carrying conductor sometimes you
will have to find out the magnetic force
on a particular charged particle so
please keep all these things in your
mind so that it becomes easy for you to
evaluate the direction at that time now
let us suppose it does now be considered
till now we consider that the conductor
is a straight current carrying conductor
which has a particular length L and a
cross sectional area is so it was a nice
conductor now let us suppose instead of
that kind of a nice conductor if you
have any conductor of some arbitrary
shape in that case how would you find
the magnetic force on that conductor so
in that case we can write let us suppose
that the conductor is of any arbitrary
shape like this like that or anything it
can be just any shape so how will you
find it in that case in that case what
we do we take a small element of the
conductor because we don't know the
shape of the conductor at all so we
cannot tell anything about it dimensions
so what we do we take a small element of
the conductor later suppose I took a
small element say DL of the conductor
and then I find out the magnetic force
on this small element of length DL of
this conductor and then I say that the
total magnetic force will be summation
of I into DL cross D or in case of
continuity we say that this way
the integration of AI DL cross B so this
is an important approach to be followed
because most of the times we will come
across conductors which will be of some
arbitrary shape via which of which we
are not sure so we will take a small
element of the conductor we will find
out the magnetic field or the magnetic
force at that small element considering
that is an element of length DL and then
we will find out the total magnetic
force on the conductor by integrating
the magnetic force experienced by that
small element alright okay now please
make a note here that the magnetic field
which we are talking about here that is
this magnetic force I L cross D here the
magnetic field which we are talking
about this magnetic field is the
external magnetic field this magnetic
field is not produced by this current
carrying conductor right I think I have
already mentioned that before like the
magnetic field is produced by some other
source and this current carrying
conductor enters the magnetic field so
this current carrying conductor
experiences the force due to magnetic
field but this magnetic field is not
reduced by this current value conductor
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