in this example we’re given that a thin
non conducting ring of mass m, radius ay and
uniformly distributed charge q, is placed
inside a solenoid magnetic field, with its
plane normal to b vector, and coaxially with
the solenoid as we can see. here we can see
this dotted line represents the solenoid boundary,
in which a magnetic induction exist along
the axis and, circular ring of radius ay and
charge q is placed. here we’re given that
if b is changing with time as b is equal to
k t. we’re required to find the angular
speed attained by the ring after time t. in
this situation we know as magnetic field is
continuously changing with time it is increasing
in inward direction. so here an anti clock
wise electric line of force will be developed
and at every point, the electric field will
be tangential to the ring. and here, we can
write, magnitude of, induced electric field,
at sircumference, of ring is, this can be
given as e is equal to, half, ay, d b by d
t and in this situation the value of d b by
d t is k. so it’ll be half ay k. that is
the electric field induced at the circumference,
and here we can directly write, torque on
ring is, due to this electric field tau can
be written as, q e, ay. this can be also written
as, half ay k is the electric field so it
is half q, ay square k, that is the torque
acting, on the ring, due to which it’ll
have some angular acceleration. and using
the torque we can directly write, angular
acceleration of ring, which can be given as
alpha which can be written as, torque upon
its moment of inertia. and for ring moment
of inertia we can use as, m ay square. so
in this situation we can see, the value of
angular acceleration we get, q k by, 2 m.
because in this situation ay square gets cancelled
out. and angular speed, attained, by ring
after time t omega can be written as alpha
t because initially, we consider the ring
to be at rest. so this’ll be q k t by, 2
m. that’ll be the answer to this problem.
