in this example, in figure a wire ring of
radius r is in pure rolling on a surface.
we’re required to find the e m f induced
across the top and bottom points, of the ring
at any instant of time. now in this situation
if we just have a look, we can directly state,
that a semi circular loop on left side and
semi circular on right hand side, at any instant
it is rotating with angular speed omega. we
can say at any instant the center of mass
is moving with a speed v, and speed of center
of mass we can write as, r omega. so for any
random shaped wire, or a semi circular wire,
we can say that, this is equivalent to a rod
of length 2 r, which is rotating about the
bottom most point because this is the point
which we can treat as instantaneous axis of
rotation of rod. and for this ring, we can
apply the direct relation of a rotating rod,
or directly a moving rod with a speed v. so
we can directly write, the motional e m f,
say these points are ay, and p, p is the bottom
most point and ay is the top most point, we
can write motional e m f induced, across points,
ay and p is, here we can see as the free electrons
of the wire moving toward right, they’ll
experience magnetic force in downward direction.
so top most point will be the high potential
point, the bottom most will be the low potential
point. so the e m f we can write as, b v l,
v we can write as, r omega, and length we
can write as 2 r. so the motional e m f we’re
getting as 2 b omega, r square that’ll be
the answer to this problem. and this can also
be handled by using, motional e m f of a rotating
rod that is half, b omega l square, in the
situation when we can treat this ring to be
replaced by a vertical rod of length 2 r hinged
at the bottom most point. so e m f we can
write as half b omega, length can be taken
as 2 r, whole square. on simplifying we’re
getting the same result 2 b omega r square.
either way we can handle the problem either
by using this relation or by using this relation.
both are valid in this situation.
