in this example we’re given that a long
solenoid has n turns per meter, radius r,
and carries a current i, is in gravity free
region. and from its axis at a distance twice
that of its radius, a charge plus q and mass
m is placed. it is saying that if the current
in solenoid, is suddenly switched off, then
we’re required to find the velocity attained
by the charge. in this situation let us first
draw the situation, say this is a solenoid,
in which due to current, say a magnetic induction
exist in inward direction, parallel to the
axis of this solenoid, and it is of radius
r. and it is saying from the solenoid, say
at a distance 2 r from the center, a charge
plus q which is of mass m is placed. now in
this situation, if we just have a look on
the magnetic induction we know magnetic induction,
inside the solenoid is, this can be directly
written as, mu not, n i. and we can see if
the magnetic induction is switched off, this
is the initial magnetic induction and final
magnetic induction will become zero. so in
this situation we can say, there’ll be a
sudden change in flux in the solenoid, due
to which at a distance 2 r from the center
we can see, an electric line of force will
be induced, which is of radius 2 r. and as
an inward direction magnetic field suddenly
decreases, it produces a current, or an induced
electric field in clock wise manner, so as
to oppose the downward dic rement in magnetic
induction. so at the location of charge, suddenly
a tangential electric field e is induced.
and this electric field we can say will last
only for that small duration, in which the
magnetic field drops from, mu not n i to zero.
and say the duration is delta t. then in this
situation we can say, electric field induced,
or we can write average electric field induced,
in duration, delta t, that is in which the
field is switched off, this can be written
as, for a region, outside the magnetic induction
we can use electric field as half, r square
by x, d b by d t, then in this situation the
value of x is 2 r. so we can write it r by
4 because when we put x is equal to 2 r, 1
r gets cancelled out, and here d b by d t
we can write as, mu not, n i divided by, delta
t or mu not n I minus zero divided by delta
t. so if this is the electric field which
lasts for a duration delta t, then we can
write, impulse, on plus q is, j we can write
as, q e delta t. and this’ll impart a velocity
v. so here, momentum attained by the charge
is m v. so in this situation we can directly
write, speed attained, by charge is, v we
can write as, q e, delta t by m. and the value
of e delta t we can substitute as, q by m
multiplied by, e delta t we can write as,
mu not, n i, r by 4, so the final result we
get is mu not, n i, r q divided by, 4 m. that’ll
be the answer to this problem.
