let us now study the magnetic force, on a
random shaped current carrying wire. here
you can see this is a, wire which carries
a current i. and the wire edges are, ay and
b. you can see the separation between the
2 end points of the wire is l. say we are
require to find the, magnetic force on this
current carrying wire if the magnetic induction
is b which is acting into the plane of the
surface. so in this situation to find out
the net magnetic force. we consider 2. axis,
1 is x axis along the line l. and another
is y axis which is perpendicular to this.
now if we consider a small element of length
d-l in this wire which carries the same current
i. to find the net force on the wire first
we resolve this element into 2 perpendicular
components one is along the length. of element
along x axis which is d-x. and another is
perpendicular to the. length of x axis which
d-y. now in this situation. if we talk about
the net magnetic force acting on this element
of length d-x it’ll be in upward direction
that is along y direction by righthand palm
rule we can see. this d-f in y direction,
if we calculate the magnetic force. on element.
d-x is. this can be written as d-f y and the
magnitude of this can be given by b-i d-x.
similarly on this element of length d-y the
magnetic force will be in x direction this
can be written as d-f-x. so, here we can write
magnetic force. on. element d-y is. this can
be given as d-f x which is written as b i
d-y. if we calculate the total magnetic force
here we can also get this d-f x and d-f-y
as vertical and horizontal component of this
force d-f which is acting on the element of
length d-l. so here total force. on wire this
we can calculate force in x direction as.
integration of d-f x. which is integration
of, b i d-y. if we wish to integrate it from
point ay to b then, y coordinate limits will
be from zero to zero so total force will come
out to be zero. similarly we calculate force
in y direction it is integration of d-f-y.
which is integration of b-i d-x. which is
integrated from zero to l from ay to b and
on integrating as b and i constant this will
give us b-i-l. so we can write, net force
on wire is. f-wire we can write as b-i-l.
and this is the same force, which would be
acting on the wire if. this curved wire is
replaced by a straight wire along, of length.
which is equal to the line joining the 2 end
points of the wire. so you can always keep
in mind that the net force acting on a random
shaped wire is equal to. the force which will
act if it is replace by a straight wire which
is equal to the. length of line joining the
2 end points of the wire always keep in mind.
so this is the way how we calculate the net
magnetic force on a random shaped wire.
