now let us study about stable and unstable
iquilibrium of a current carrying loop in
magnetic field. here you can see there are,
2 square or rectangular shaped coil placed
in, uniform magnetic field b. in 1 coil a-b-c-d
the current is flowing in clockwise manner
and the other coil p-q-r-s current i is flowing
in anticlockwise manner. now in this situation
if we analyze this first case here we can
see. as for the direction of this current
carrying loop its magnetic moment can be taken
in inward direction. so here its m vector
which we define as current multiplied by the
enclosed area it exists in inward direction.
that is parallel to the direction of magnetic
field. so we write here. m vector and b vector.
are parallel. and angle between them can be
taken as zero degree. so in this situation
if we calculate the torque. on loop. and this
torque we write as, m cross b. so in this
situation, it’ll be m b sine-theta. so this
can be taken as zero. if there is no torque
acting on the loop. we can say that the loop
will be in iquilibrium. and in this situation
we can also calculate the interaction energy
of this loop. with the magnetic field. so
this can be written as minus m dot b. and,
substituting the value of angle this will
be minus m-b. that is the minimum possible
energy which is possible in this situation.
so for minimum energy we can state, that.
this is the state of. stable iquilibrium because
the interaction energy. is minimum. this we
can also define in terms of forces. like here
we can see. this wire a-b will experience
a magnetic force in upward direction by righthand
thumb rule you can check. on this wire b-c
the magnetic force is toward right. on all
these wires the magnetic forces are acting
in the direction. away from each other. in
this situation if we consider. an axis of
rotation if we consider this is axis of rotation.
and we slightly tilt this coil like, if we
tilt it by an angle theta in this manner you
can see. the magnetic forces which are acting
on sections a-d and b-c. will have a tendency
to pull the coil back to its iquilibrium position
and it will oscillate about the axis of rotation
like this. in this situation we can see the
magnetic forces. acting on wire section b-c
and a-d will create a couple. and due to which.
the coil will always. has. restoring tendency.
so we can say in this situation the coil or
this current carrying loop will be in stable
iquilibrium. similar situation if we analyze
in this situation where we can see. according
to the direction of currents its magnetic
moment. m vector exist in outward direction
which is along the area vector given by the
circulation of current. so here we can write.
angle between. m vector and b vector is, this
theta is 1 80 degree or pie radians. so again
in this situation we can see. torque on loop,
which is given as tau is m cross b. here again
we can see as the angle is 1 80 degree the
net torque acting on the loop will be zero
so this is also state of iquilibrium. but
here if we calculate the interaction energy
of the loop with the magnetic induction it’ll
be minus m dot b. and i am substituting the
value of theta 1 80 degree as minus m b coz
theta it’ll be m-b. which we can see the
maximum possible energy in this situation.
so this must be the state of. unstable iquilibrium.
and this we can analyze by the direction of
forces again. like here. like previous case
we consider an axis of rotation. the direction
of magnetic forces on wire. q-r we can see.
this will be in inward direction. this is
the magnetic force on q-r and, on p-s. it’ll
be in, rightward direction. this is also inward.
so for these magnetic forces we can see if
the coil is slightly tilted. by an angle theta,
with the, direction of magnetic forces you
can see. the couple produce by these forces
will continue to rotate the coil till it will
reach. the, final position similar to this
which is the stable iquilibrium position and
it’ll start oscillating about that point.
so here we can see if we slightly tilt the
coil. the couple of magnetic force. will exert
a torque, on the coil. in the direction away
from this iquilibrium position which also
verify that the state of iquilibrium is unstable
in nature.
