in this example. we are required to find the
magnetic moment of the current carrying loop
shown in figure. here if we just have a look
on the figure we can see this is three dimensional
loop. in which these 2 wires segments are
in x-y plane. making a square of edge length
r. this is a quarter circular arc in y-z plane.
and this is straight wire. in x-z plane. so
to find out the magnetic moment of this loop
it is advisable that in such situations. to
split the loop in. 3 loops in 3 different
planes like . you just have a look on the.
alternative situation i am drawing here. this
single loop I can split into 3 loops like
this-this is a square loop. which is carrying
a current i in anticlockwise manner. another
is a quarter circular. segment. which is also
carrying a current in anticlockwise manner.
and another is a triangular, loop. which is
carrying a current, in this direction. anticlockwise
manner. and here you can see the 3 loops,
are placed edge to edge along with the, 3
axis. of this coordinate system you can see.
the wires carrying current along the length
of the axis, will cancel out there currents
and only the outer loop will be left over.
so here we can see for this loop, which is
having a magnetic moment. i multiplied by
r square you can see its magnetic moment is
in the direction of z axis. this is the, quarter
circular, segment which is having area pie
r square by 4. if we just have a look on righthand
thumb rule we can see its magnetic moment
is in x direction. and for this triangular
loop. the current is in anticlockwise manner
and here the magnetic moment will be in the
direction of y axis. so here for all the 3
we need to add up because adding up the 3
loops we are getting the original loop. so
in this situation, for this quarter circle
we can write magnetic moment is current multiplied
by pie r square by 4. and its magnetic moment
is in x direction so it is i-cap plus. for
this triangular loop the total area will be
r square by 2. so magnetic moment will be
i r square by 2 j cap. plus for this square
loop the magnetic moment is i multiplied by
r square and the direction is k cap. so this
will be the answer to this problem that is
the total magnetic moment of this 3 dimensional
loop. which we have calculated by splitting
it into 3. independent loops. carrying current
in similar directions. with. there, edges,
along the axis of coordinates system will
be having opposite and equal current so that
when the 3 loops are joint edge to edge along
the axis. of this coordinate system, the original
loop will be formed. so this is method which
we are going to use in many cases. and whenever
there is a loop which is, 3 dimensional. and
it is advisable to split it into. 2 or more
loops.
