in this example, we’re given a rod ay b
of length L is placed in a cylindrical region
time varying magnetic field as shown in figure,
with its rate of increasing c tesla per second,
it is the value of d b by d t. and we’re
required to find the potential difference
across the ends of the rod due to induced
electric field in the region. now in this
situation, if we just analyze, the potential
difference across ends ay and b, then we can
see at every point of the rod, the distance
from center of this region is different, so
induced electric field will also be different.
so to calculate it, let us consider an element
of width d x from the center of the rod, and
at this point we can see as, in downward direction
magnetic induction is increasing, and electric
field is induced in anti clock wise manner,
so at the location of this d x, electric field
will be tangential to, the circular electric
line of force, so this’ll be in this direction.
say if this angle is theta, correspondingly
we can say this angle is also theta. so we
can write, in this situation, potential difference,
across the element, d x is, it’ll be d e,
this we can write as, e d x coz theta, because,
across 2 points where electric field is e
we can write, the potential difference d v
is e dot d x, so it’ll be e d x coz-theta.
and value of electric field inside at a distance,
this is root of r square plus x square, the
electric field we can write as half root of,
r square plus x square multiplied by, d v
by d t which is c here, multiplied by d x,
multiplied by, we put the value of coz-theta
from this triangle it’ll be r divided by,
root of r square plus x square, which gets
cancelled out, and here in this situation
the value of e m f induced across the element
d x we can write as, half c r, d x. so we
can directly state total potential difference,
across the rod, ay b is, this can be given
as integration of d e which is integrated,
within limits from minus l by 2 to plus l
by 2. so value of e m f we get as, half, c
r, l, that’ll be the answer to this problem.
but in this situation if we have a look on,
the high and low potential end we can see
that, e coz-theta is toward point b, but in
this situation 1 important thing which useful
is, when we talk about the free electrons
of the rod, we can see as, induced electric
field is toward right, the free electrons
in the rod will experience a force toward
left and will drift to point ay, so point
ay will be slightly negative and due to shift
of electrons b will be slightly positive.
so we can directly state in this situation,
point ay will be the low potential end, and
point b will be the high potential end. you
must be very careful about, the polarity of
the potential difference induced across point
ay and b, as due to induced electric field
we can see it exist from ay to b, so conventionally
b should be at low potential but here, as
the potential is developed due to shift of
the charges, the polarity will be opposite
in this situation.
