now let us study about induced electric field,
outside a cylindrical region time varying
magnetic field. to understand the situation
let’s have a look on this picture, here
we’re given with a long solenoid in which
a current i is flowing, which produces a magnetic
induction b along the axis of this solenoid.
and outside the solenoid there’s a coil
along with a galvanometer. and, here if this
current increases with time, then this magnetic
field will also increase with time, and on
changing this magnetic field we can say the
flux passing through the outer coil, is also
changing. due to which we can say the galvanometer
will be deflected as a current is induced
in the outer coil. that means, in this outer
coil if magnetic flux in this direction is
increasing, this’ll have a tendency to induce
a current, in anti clock wise manner so as
to oppose the increase in this magnetic field
and the galvanometer will deflect on the left
hand side. so we can say, this induced current,
in the outer coil is also induced by some
electric field at this point. so we can say
whenever in a region magnetic field is changing,
outside the magnetic field also, tangential
to the coil, or a circular path which is symmetric
with respect to the cylindrical region and
electric field is induced. and again the line
of forces of this electric field will be,
circular. and let’s have a look on the situation
in a different way, like if we just have a
look on the cross section of this, it’ll
look like this picture. in this picture we
can see, we’ve drawn this as a cross section
of solenoid in which magnetic induction is
in inward direction which is of radius r.
and at a distance x outside, we consider a
circular path, we’ve removed the outer coil
and considered a path, and as magnetic field
is increasing in inward direction, an anti
clock wise electric field must be induced
over here, of which the direction at every
point will be tangential to this circular
path. and if their exist a coil this electric
field will give rise to an induced current
in the coil. on this situation we can directly
state, for outer points, that is for x greater
than r, here we can use, the total e m f induced
by faraday’s law we can write, e is pi r
square d b by d t and, it’ll always remain
constant no matter how large the path is,
because no matter how far we move away from
center total flux remain same. and it must
be equal to integration of, e dot d l, for
the closed path, outside the field. so it’ll
be integrated from zero to 2 pi x. so in this
situation, we get as, symmetrically we can
say, e is uniform and can be taken out of
integration of d l, from zero to 2 pi x will
be equal to pi r square d b by d t. and further
we can write it as, e into 2 pi x, which is
equal to pi r square d b by d t. and in this
situation pi gets cancelled out and the magnitude
of electric field we get as, half, r square
by x, d b by d t, that means in this situation
we can see, the magnitude of electric field
decreases as we move away from this center.
so in previous section we’ve studied about,
the orientation or configuration of electric
lines inside, and we’ve seen as we move
away from center inside the density of electric
lines, increases. but here, outside points,
for x greater than r, for outside points electric
field is decreasing with x. so here we can
say, as we move away, the density of electric
lines will start decreasing or we can say
the line seperation will increase as we move
away from, the center outside the cylindrical
region. so be careful about the magnitude
as well as, the orientation of electric lines
in the outside region.
