Moving on to our next
problem,here is a waveform of
current passing through inductor
of resistance 1 ohm and
inductance 2 Henry, we need to
find the energy absorbed by the
inductor in first 4 second, it
is saying that there is a
inductance of resistance 1 ohm,
actually the practical model of
inductor is like this, every
inductor is nothing but a coil.
So, coil will have some
resistance, it means it is not a
pure inductor, this inductor is
made up of coil and the coil is
having resistance, it means
there will be some losses in
this resistance, it is an impure
inductor okay. So, it is saying
we need to find energy absorbed
by inductor in first 4 seconds.
So, there will be two component
of energy here one is the energy
that is being dissipated in the
resistor as a heat energy and
other is the energy that is
being absorbed or stored by the
inductor. So, we will find both
the energies and the total of
energies will give me the total
energy absorbed by the inductor.
So, let us first find the energy
stored in the inductance okay.
We know that voltage in the
inductor is given V is equals to
Ldi by dt, we can find out di by
dt okay because current waveform
is given here, so in the first
zero to two second di by dt is
the slope, the slope of this
waveform is that is six by two
okay six divided by two that is
three ampere per second, this
three ampere per second is the
di by dt, so value of di by dt
it is three and the value of
inductor is two. So, the voltage
will come out to be two into
three that is six volt and that
is for the duration zero to two
second Okay, after two second
you can see here current
waveform is flat that means di
by dt is zero slope is zero,
that is zero ampere per second.
So voltage will also be zero
here, VL is zero for t greater
than zero. So the waveform of
voltage will look like this.
Okay, zero to two seconds it is
six volt, you can see here it is
six volt, after two seconds it
is zero. Now power I know power
is equals to voltage into
current. So if you do voltage
into current, we know that our
After two seconds voltage goes
to zero. So, that product will
also go to zero. So, if you do
voltage into current we will get
a waveform like this. after two
seconds you can see here, power
is getting zero because V into
I, V is zero after two second.
So, that is why power is also
getting zero. At time t is equal
to zero you can see our voltage
is six current is zero. So,
power is also getting zero okay
at time t is equals to 2 second
you can see voltage is six volt
and current is also six ampere,
two six into six 36. So, you can
see at a time is equal 2 second
power is 36. So, this is our
power waveform of the inductor.
Now energy is given by integral
of Pdt, what is Pdt what is
integral pdt, that is area under
the power curve okay this area
if you calculate this area we
will get the energy okay. This
is basically a triangle so the
area will come out to be hop
base into height, base is 2,  I
have written 2 here and the
height is 36. So we get the
energy that is 36 joule. So 36
joules the energy that is stored
in the inductance. Now we need
to find the energy that is being
dissipated in the resistance. So
we know current waveform is
something like this, okay. So
power dissipated in the
resistance is given by, I square
into R. So, from this waveform
we can find the expression of I,
that is, current is equals to
3t, you can see, equation of
this current waveform is 3t, at
time t is equal to zero, current
is zero, at time t is equal to
two second current is six
ampere, put the value of t is
equal to zero you will get zero
at a time t equal to two you
will get six. So, this is the
equation of current from zero to
two seconds. After two second
the value of current is six
ampere that is current is six
ampere for t greater than two
Okay. Now we can calculate the
energy that is energy is pdt,
power is given by i squared r,
we will get 3t square, Where
from zero to two second i
squared that is 3t, i square
into r dt okay pdt that is, plus
from two to four, current is 6
so 6 square into R dt Okay. Now,
the integral of nine t squared
is nine t square and the value
of r is one. So, integration of
nine t square is 9 t cube by
three and the limit is from zero
to two seconds okay. And the
value of r is one, plus integral
of 36 is 36t and the limit is
from two to four second okay.
after substituting the limit you
will get you will get from here
you will get 24 and from here
you will get 72 Okay. The total
energy dissipated in the
resistance is coming out to be
96 joules that is the amount of
energy which is being dissipated
in the resistor so  the total
energy would be 96 plus 36.  96
is the energy that has been
distributed in the register and
36 is the energy that is being
stored in the inductor So, 132
joule is the total energy
absorbed by the inductor from
the source out of which 96 Joule
is being dissipated as a heat
and 36 Joule is being stored in
he inductor . Okay?
