let us discuss about motional e m f under
external force and power transfer due to the
force. here we can see we’re given with,
2 smooth rails on which, a rod p q is placed.
let us consider the mass of this rod is m.
and the rails are, in a uniform magnetic field
induction b which exist in inward direction
and the 2 rails are connected with a resistance
r. now in this situation if at t equal to
zero the rod is applied, by a force f. the
rod will start accelerating and, when it’ll
attain a speed v, we can directly write motional
e m f, in rod, when its, velocity is v. this
can be directly written as b v l. and in this
situation, we can see by right hand palm rule,
the free electrons of the rod will experience
downward force, so the upper end is positive
and the lower end is negative. so it’ll
induce a current i in the resistance connected.
so we can write, current in circuit, can be
directly written as, b v l by, r. and this’ll
be the current in the rod also, and as the
rod is moving, this current will lead to a
magnetic force on rod towards leftward direction
and this’ll be equal to b i l. so here we
can say if, ay is acceleration of rod, we
can write the value of acceleration will be,
f minus b i l by, m. and now if we substitute
the value of current, this’ll be b v l by
r, so this’ll be f r minus, b square l square
v by, m r. and here we can write acceleration
is d v by d t. and we can seperate the variables,
this can be written as, d v by f r minus,
b square, l square v, is equal to d t by m
r. now in this situation we can integrate
this expression to find the velocity of rod
as a function of time. it’ll be integrated
from zero to t and, speed will be from zero
to v. on integrating we get it as ellen of,
f r minus, b square, l square v, and we apply
limits from zero to v and this’ll be t by,
m r. and here we’re having a term minus
1 by b square, l square. on substituting the
limits we get, ellen of, f r minus, b square,
l square v by, f r, it is equal to negative
of, b square, l square t by, m r. and if we
further simplify this expression, we can get,
by taking anti log on both the sides, this’ll
be f r minus, b square, l square v, by f r
equal to e to power, minus b square, l square
t, by m r. let’s continue on the next sheet
the further analysis.
now in continuation if we further simplify
the expression we can directly get the value
of speed of rod as a function of time, and
here this can be given as, f r by, b square,
l square, 1 minus e to power minus, b square
l square t by, m r. this is the velocity of
rod as a function of time and here we can
see this is an exponential function. and if
this is a velocity attained due to external
force f, we can write, power supplied, by
external force, it is p, which can be given
as, f v,and on putting the value this’ll
be f square r by, b square l square, 1 minus
e to power, minus b square, l square t by,
m r. so this is the way how power is being
applied by the external force the rate of
supply of energy, and this situation we can
see, at t tending to infinity, that is after
a long time if we just have a look on this
situation, we can see that speed will approach
to the value, f r by b square, l square, which
will become a constant. and this constant
speed we call, terminal speed, or we can say
the state is called as steady state when the
terminal speed is attained by the rod, and
this is attained because as speed increases,
if we just have a look on the rod, external
force is acting toward right and the magnetic
force b i l is acting to left. so as its speed
increases the current in the rod also increases
and b i l increases. once the value of b i
l becomes equal to f, we can directly state
the acceleration of conductor becomes zero,
and v will approach to a constant value that
is, what we call terminal speed. this terminal
speed can also be obtained, when the magnetic
force balances the external force. so we can
say, when b i l is equal to f, and the value
of i we write as b l v by r, it’ll be b
square, l square, v by r is, f. we can see
this expression will also lead us to the velocity
which is given as, f r by b square, l square.
so here we can say when steady state is achieved
a terminal speed is attained by the rod. we
can directly write the current in conductor
to be equal to, f upon b l. and if the current
also becomes a constant, if we calculate,
the thermal power produced, in resistance,
thermal power can be written as i square r,
substituting the values we get f square r,
by b square, l square. and if this is the
terminal speed we can also write, at terminal
speed, power supplied, by external force,
can be directly given as, supplied power is
equal to, f multiplied by the terminal speed
which is f r by, b square l square. we can
see it is f square r by, b square, l square.
so here we can see in this situation the 2
powers are equal, that means after steady
state is achieved a terminal speed is attained
by the rod. whatever power is spend by the
external agent, will be exactly dissipated
in the resistance as thermal power. so this
thermal power can also be obtained by substituting
current as b v l, by r, so if we put the value
of current this’ll be, b square l square,
v square by r, this is alternative expression
we can use for power supplied or, thermal
power in steady state after terminal speed
is attained. this analysis is quite useful
and, will be the basis of many different kind
of problems. so you must remember, this analysis
of motional e m f under influence of an external
force.
