in this example, here the figure shows a small
circular coil of area ay, suspended from a
point o by a string of length l, in a uniform
magnetic induction b in horizontal direction.
it is saying if the coil is set into oscillations
like a simple pendulum, by displacing it a
small angle-theta not as we can see. we’re
required to find the induced e m f in the
coil as a function of time. we’re also required
to azume the plane of coil is always in the
plane of string. so in this situation we can
see, when we release the coil, it’ll start
oscillating like a pendulum, of which the
angular frequency of oscillation omega we
know it is, root of, g by l. time period will
be 2 pi root of l by g. in this situation
if at t equal to zero it is released, at any
time we can see, the ring will be, at a position
making an angle-theta with the vertical, and
this angle-theta we can directly write by
using the s echch m equation. we can see after
time t, at instant t, the angular position,
of coil, from vertical is, this is given as
theta, is equal to theta-not coz of omega
t. and from this expression we can also find
out the angular speed of, rotation of this
rod that is d-theta by d t, this can be given
as, theta-not omega sine of, omega t, we also
put a negative sign by derivative. this is
denoting that in this situation, the coil
is coming toward the mean position, it is
giving us the direction. and if we talk about
at time t, magnetic flux, through the coil,
this magnetic flux we can write as b dot ay
the magnetic induction is toward right, and
its area vector we can say, this’ll be in
a direction perpendicular to its surface.
if this angle is theta, the angle which area
vector is making with magnetic field vector
is also theta. so this can be written as b
ay, coz-theta, which can be further written
as, b ay coz of, here the value of-theta we
can write as theta-not coz omega t, so it’ll
be theta-not coz of, root of g by l t, if
we’re required to find the magnetic flux,
as a function of time, this is the result,
to find out the e m f induced in the rod,
we can use faraday’s law. let’s continue
on the next sheet.
as we’ve seen the magnetic flux passing
through the coil is given as b ay coz-theta
we can write, e m f induced, in coil is, it
is mod of, d phi by d t. on differentiating
we get it as, b ay, sine-theta d-theta by
d t. in this situation, for small theta, we
can write sine-theta is approximately equal
to theta, so here e m f induced can be written
as b ay theta, d-theta by d t. if we substitute
the values, e m f that we get is b ay, value
of-theta is, theta-not coz of, root g by l,
t, and in this situation d-theta by d t, as
we already calculated in magnitude, it is
theta-not omega, sine of, root of g by l t.
now in this situation we can further simplify
the result, e m f here, coz root g by l t,
sine root g by l t we can write, two sine
of 2 root l t by 2. so it’ll be half, b
ay, theta-not square, omega, sine of twice
of, root g by l into t. this’ll be the answer
to this problem, that is the e m f induced
in the coil as a function of time.
