Diapole in uniform electric field - Electric Charges & Fields Video Lecture - Class 12

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before going ahead with part 27 we will
now study what happens when an electric
dipole is placed in a uniform in
external electric field that means there
is an electric field which is already
present on top of that we place an
electric dipole inside the electric
field so will there something will there
be something unusual happening in this
case something new will happen here let
us have a look dipole in a uniform
electric field experiences talk oh so a
new term negative talk I am sure that
all of you are aware of waters torque
torque is nothing but it is force in
rotational motion that means it is kind
of angular force a force which is
applied to an object in certain angle
that makes the object rotate itself so
that means this statements mean to say
that if a dipole is placed in a uniform
electric field torque will make it
rotate is that possible if yes how let
us see let us suppose this is uniform
electric field the blue lines represent
represents uniform electric field right
and let us suppose that these two small
red dots represent a dipole let us say
this is minus Q and this is plus Q so if
this dipole is placed in a uniform
electric field randomly making some
angle theta with a electric field let us
suppose it makes angle theta with the
direction of the electric field and the
length of the dipole is 2a that is
distance between the two charges is - it
so what will happen to this dipole now
let us calculate the force on the force
that will act on each of these charge
let us mark this point as a and let us
mark this point as B so first let us
calculate the force which will act on a
so what could be the force that will act
on a the force will act due to the
electric field so what is the force that
will arise due to electric field
electric field e multiplied by the
charge so that is DQ in which direction
in the direction of electric field
because this is a positive charge and
the electric field moves from positive
to negative this will be the direction
of electric field electric field always
moves from positive towards negative so
the electric field will push this charge
towards this side because here it is
negative so positive will attract to the
negative so the direction will be this
side right so the force on a will act
along this direction so Q e in the
direction of e in the direction of the
electric field
E now let us see what is the force on
the force on B will also be equal to e
into Q but since this is the negative
charge so this will be opposite to e so
the direction would be opposite to e so
that means what do we see we see that
two equal forces are acting along the
two ends of the dipole but in opposite
direction so that means let us suppose
forget about this scenario let us
suppose you have a small stick you apply
a force F at one the end point at one
end point in this direction and you
apply us the same force F at the other
end point in the opposite direction so
what will happen this will make this rod
rotate like this so the rod will start
rotating like this so it will rotate
clockwise imagine just hold the pen in
your hand apply a force at one end in
this direction and a force the same for
from this direction you will see that
the paint pen is rotating so when two
equal and opposite forces are applied to
an object at different lines of force
then that object rotates so observe an
important thing here let us suppose if
you have if you f there is a point here
you apply a force F in this direction
you apply an equal and opposite force in
this direction what will happen to this
point it will not move because the net
force is equal to zero but what happens
if you have a rod and you apply a force
F here and equal and opposite force at
the other end here also the net force is
zero because the net force here is equal
and opposite but here since the line of
force is different therefore the body
starts rotating and we see that the body
has a torque right so therefore in this
case also the similar thing happens the
dipole starts rotating and the dipole
experiences a torque now we have to
calculate so here we see net force is
equal to zero which force
translating force so net translating
force is equal to zero but torque is
produced because equal and opposite
forces are applied but along different
lines of force so how do we calculate
torque torque is nothing but force
applied into the displacement so here
what is the force applied force applied
let us suppose is f into the
displacement between displacement in the
sense the perpendicular distance between
the two dipoles so the perpendicular
distance is given by this length so what
is this length if you look at this if
you look at this triangle you can find
that this length is let us call this
point as C so this will be F into AC so
what is F F is nothing but e into Q and
AC is nothing but 2a
sine-theta looking at this right angle
triangle because sine theta is equal to
perpendicular that is AC divided by
hypotenuse that is to it
so this will be equal to now Q that is
charged into the distance between them
is nothing but dipole moment so PE sine
theta so therefore we can say that
torque can be written as cross-product
of dipole moment and that electric field
so torque is equal to t cross e so what
we observe the new thing here that a
dipole when placed in a uniform electric
field it experiences a torque and the
torque is defined as the cross product
of the dipole moment vector and the
electric field vector so now let us
discuss a few points on the top
experienced by experienced by the dipole
in uniform electric field so when do you
think the torque will be maximum just
now we derived the expression torque is
equal to P cross e or we can say torque
is equal to P e sine theta so if I want
the thought to be maximum this quantity
should have maximum value now the
maximum value of sine theta is 1 so
thought will be maximum when sine theta
is equal to 1 right
therefore torque maximum will be equal
to PE this is the value of maximum
torque and when torque is maximum your
sine theta is equal to 1 right now
when will the talk be minimum now
similarly using this we can say what is
the minimum value of sine theta to 0 so
the minimum value of torque will
definitely be 0 what happens if theta is
very small let us suppose that the
dipole let us suppose this is the
electric field
and the dipole is aligned in the
electric field some somewhat like this
that is the dipole also appears to be
almost straight but it makes a small
angle with a liquid feet so what happens
in that is what is the torque
experienced in that case we say that the
torque is equal to PE sine theta since
theta is very small so sine theta can be
replaced with theta so we say that
torque is equal to PE theta also we know
that by definition of torque
we know that torque is also defined as
product of moment of inertia and angular
acceleration you would have studied this
in your rotational motion chapter in
class xi that torque is defined as the
product of moment of inertia and the
angular acceleration so if we can equate
these two so thought is equal to PE
theta and torque is also equal to I into
beta so if you equate 1 and 2 you can
equate the RHS of 1 and 2 so you get IB
done is equal to PE theta so from this
you can say beta is equal to PE theta
divided by I what is beta beta is
nothing but angular acceleration let me
write it down here this is angular
acceleration so angular acceleration can
be written as Omega square into theta
where Omega is angular velocity and
theta is angular displacement so this is
equal to PE theta divided by I so this
theta and theta will get cancelled so
Omega will be equal to root over PE
divided by I or you can say that time
period of the rotation T which is equal
to 2 pi by Omega can be written as 2 pi
root over I divided by t e don't think I
went
of the track because since I was
discussing about talk I think this was
also an important thing people because
many times you get problems like this
that there is an electric dipole placed
in an electric field now of so-and-so
magnitude and then it starts rotating so
what is the time period of rotation you
can get questions like that so I just
discussed this so that you get an idea
to how to approach such kind of
situations right so this way we can find
out the time period so what did you
observe then net observation was that
that the net force in this case is zero
however the net torque is not equal to
zero now whenever to talk of uniform
electric field till now we discussed
uniform electric field so what happened
in case of uniform electric field in
uniform electric field the net force
acting on the dipole is equal to zero
but the net torque acting on the dipole
is not zero now what would happen if the
uniform electric field is replaced with
a non uniform electric field in case of
a non uniform electric field the net
force is also not equal to zero and the
net torque is also not equal to zero so
in case of a non-uniform electric field
the net force is also not equal to zero
the forces are not equal and opposite
right so this was all about the torque
experienced by a dipole when placed in a
uniform electric field thank you please
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