let us discuss the induced electric field
inside a cylindrical region in time varying
magnetic field. here we can see if this is
a cylindrical region like a solenoid, inside
which a magnetic induction b is there, and
say if b starts increasing with time, if there
would have been a coil placed inside, an anti
clock wise current would have been induced.
but even if there’s no coil, we can say
that inside at a distance x, a circular electric
line of force will be induced in anti clock
wise manner, corresponding to the electric
field which is induced at every point, at
a distance x from center in such a way, that
direction of electric field is tangential
to this circular line. and we can directly
write, for the closed path, of a radius x.
in this situation for the closed path of radius
x, the total e m f induced, if we would have
placed a coil over here, then total e m f
we write as pi x square d b by d t. and the
total work done in circulating a unit positive
charge round the loop we can write it as integration
of, e dot d l, for the closed path from zero
to 2 pi x. for this closed path the total
work done, must be equal to the e m f induced
in the loop as loop e m f is defined in a
way as, it is the potential difference across
2 points, which are adjoining of the same
coil, or it is work done in a unit positive
charge going round the coil. so in this situation
as at every point symmetrically, e and d l
are parallel, we can take this e out of this
sign of integration it is integration of dl
from zero to 2 pi x, this’ll be pi x square
d b by d t. and further we can write it as,
e into, 2 pi x is, pi x square d b by d t.
here, pi gets cancelled out, x also gets cancelled.
so the magnitude of induced electric field
here is half x d b by d t, this is for all
points we can write, for x less than r. that
means inside the cylindrical region the electric
field is directly proportional to the distance
from center. so as we move away from the center
of cylindrical region, the electric line of
force gets denser and denser, which we’ve
seen in the previous section. let’s have
a look on the configuration of electric line
of forces once again here. in this picture
we can see, how the density of electric lines,
are varying as we move away from the center,
from this expression we can see, as electric
field is proportional to x as we move away
from the center magnitude of induced electric
field increases, corresponding to which we
can see the density of lines also increases
as we move away from the center of this magnetic
field.
