let us now study the torque due to magnetic
forces on a current carrying loop in magnetic
field. here you can see this is a uniform
magnetic field in which a. rectangular coil,
a-b-c-d is placed which is carrying a current
i in clockwise manner. the dimensions of the
loop are given as l and b. in this situation
if we just analyze the forces acting on different
wire segments of this loop. here we can say
that. force on. wire sections. here we talk
about sections, a-b and, this section c-d.
is zero, directly we can write because here
the direction of current or the length is
along magnetic induction. and we already discussed
that those sections of wire which are parallel
to magnetic field. doe not experience any
magnetic force. now in this situation if we
calculate the magnitude of force. on the wire
sections b-c and, d-a. this can be given as
f is equal to b-i-l because magnetic induction
and. the length of wire are perpendicular
to each other, and by using righthand palm
rule we can see on the section d-a magnetic
force will experienced. by it in inward direction.
and on this section. b-c it will experience
in upward direction, outward direction. here
we can see again as a forces are opposite
net force on the loop will be zero which we
already discussed in previous sections. but
here the line of action of these forces are
not same. so. these 2 forces equal and opposite
acting on the loop will create a couple. so
here we can calculate the torque of. couple
forces. on loop is. this torque we can write
as force magnitude multiplied by the separation
between the 2 forces that is. b which is the
breadth of the loop so this can be written
here as. b, i l b. which can be written as
b i ay where, the product of l and b is the
area enclosed by the loop. here we can also
write if there are. n turns. in loop. total
torque. more precisely we should write magnetic
torque. on loop. is this can be written as
tau is equal to we just multiplied by n this
will be b i ay multiplied by n. which can
be written as, b i n-a that is total torque.
acting on the. current carrying coil which
is placed in the magnetic field. and this
can be further modified if the coil rotates
by some angle the magnitude of torque will
change like. here if we, tilt this coil by
an angle theta in this manner the separation
between the 2 forces will decrease lets continue
on the next sheet for the same. here in continuation
you can see if the coil is rotated by an angle.
such that. its area vector which is normal
to direction of this coil. makes an angle
theta with the direction of magnetic induction.
then in this situation also you can see the
force on wire a-d will be in inward direction.
and force on wire b-c is in outward direction.
the magnitude of force is same as b-i-l because
both of these wires are of length l. but you
can see the separation between the 2 wires.
or separation between the 2 line of action
of these forces, is reduced to b sine-theta.
because when the coil was in the plane of
this magnetic field. the separation was b
and when it is tilted in a way that. its.
normal to surface makes an angle theta with
b vector this line of action, is reduced to
b sine-theta. so we can write when. area vector.
of this current carrying coil. or, loop. makes
an angle. theta to. direction of magnetic
induction. then, torque of. couple. couple
of these 2 forces. on loop is given as. this
can be directly calculated as torque here
is b-i-l is the force. multiplied by the separation
which is b sine theta we also multiplied with
a number of turns in the coil. on anytime
the wires. are circulating. so here the torque
can be written as, here l multiplied by b,
can be written as area of this coil. so it’ll
be, b i n ay sine theta, that is torque acting
on it. now in this situation. this constant
term i n ay we can replace it by single constant.
m, so it can be written as m b sine theta.
where. m is written as the product of current
number of turns and area and this m is also
a vector quantity which exist along the direction
of area vector only. and its, actual direction
is given by righthand thumb rule, as current
is circulating clockwise m vector will exist.
in the direction of your thumb. i also write
it later. but here lets just keep in mind
that m which is the product of current number
of turns and area. is called. magnetic dipole
moment. or. magnetic moment. of loop. current
carrying loop. and in this situation if m
is considered as a vector vectorially. this
torque can be given as. m cross b. as. if
we just rotate our righthand from m to b you
can see a clockwise torque is acting on it
and due to this couple also, the torque acting
on the coil is in clockwise direction. and
you can also write down a note over here.
where you can say. the direction of. m vector
which is equal to i n-a vector. is always.
in the direction of. ay vector. with. right
hand thumb rule. pointing. thumb direction,
when. fingers. along circulation of, current.
that means if a current is flowing in clockwise
manner you circulate your fingers your thumb
will be pointing in inward direction so magnetic
moment will exist. along area vector which
is. pointing in the direction of thumb. otherwise
if we talk only about area vector for the
surface which is facing this side area vector
will be perpendicular to this on the other
side also. but magnetic moment will have only
1 direction so just be careful about the whole
analysis here.
