LC Oscillations Video Lecture | Physics for JEE Main & Advanced

today we take a look at a combination of two circuit elements
a capacitor and an inductor
that's all we have in this circuit now
this circuit is similar to a mass
spring system this is something you've
studied
earlier that if you put this mass here and release it
then this system will oscillate
this similarity seems strange
but you'll be surprised to see that the
parallel between the two
is exact and we are also going to write
the same differential equation for both
the cases
now if you're wondering that why is the
name
LC oscillator for this simple
circuit which is made up of
just an inductor and a capacitor
See once we discuss the working of the circuit
you can see that in this combination
that
that the charge and the potential difference
and the current in the circuit vary sinusoidally
with a time period of T and an angular frequency omega
Hence the name
LC oscillator.
Now lets take a look at this-how does this work
see initially the capacitor is charged lets say this plate carries a positive charge and this one is negative
charge
and this is connected to an inductor
see the starting point is similar to the
case where
in the mass-spring system the mass must be pulled
to one side and released and then it
would
oscillate
similarly in this case there is an exchange of
energy between the
electric field of the capacitor
and the magnetic field that builds up around the inductor
and the energy stored
in the capacitor is given by
the energy stored in the magnetic field
on paying that bill is given by hot
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is this then pinned up then and I use
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in-depth and
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and there's a draw them I'm then explain
what do they mean
you have a capacitor and an inductor
in the first case the capacitor
is fully charged and there is an
electric field
between the plates of the capacitor
and there is no magnetic field
around the inductor
the energy stored in the capacitor is
maximum and the energy
stored
in the inductor is 0.
and the sum of the two energies
is the same as this
this is the sum of the two energies so
this is the total energy
this is the energy in the electric field of
the capacitor
and this is the energy in the magnetic
field around the inductor
in the next stage
the capacitor gets fully discharged
and this is the magnetic field present around the
inductor and when this
magnetic feed around
the inductor becomes maximum at
that instance
the electric field between the plates of the capacitor is
zero because the plates are fully
discharged
and in this case the current is
maximum
so if you look at the energy in this
case the energy in the electric field
has become 0 the energy
in the magnetic field is now maximum
and the sum of the two energies is once
again the
same so this is the energy in the
electric field this the energy in the
magnetic field
and this is the total energy. Now next
the capacitor gets charged
once again and
you can see that in this case the
polarity gets reversed
so now this plate acquires a negative
charge and
and this plate is positively charged. Once
again
the electric field is maximum
so the energy in the electric field is
maximum
energy in case of the inductor is zero
and the total energy is the same. In the
next stage
we have a situation similar to the
second stage
so you have the capacitor completely
discharged
the magnetic field around the inductor
would be maximum its just that
the direction of the field lines will be
reversed now
so in this case if you draw the
the electric field and the magnetic field
then the electric field between the plates of the capacitor is zero.
the might be feed around aimed at ben is
a maximum
and the total energy is once again the
same
as it has been in all these
different cases
this is the magnetic field
and now coming to the final stage
you'll be actually back to this
you have the capacitor once again charged
so you are back to the original starting
point
and the energy distribution would be
maximum in the electric field
be 0 in the magnetic field and the total
energy would be
constant so now we are going to discuss
this
energy transfer so if you just label
this is the one
2, 3,4 and
5
see in the first case when the charged capacitor is connected to the inductor
the capacitor starts to discharge and
the inductor
and this means that the current
starts flowing into circuit see if I
were to draw a graph
this is the starting point which is 1
and then the current
starts increasing and it reaches a
maximum value at
2 so the current
becomes maximum in this case and at this
point of time
magnetic field around the inductor is
maximum
and the energy is transferred to the
magnetic field
and this now appears around the inductor, the electric field has decreased
and the magnetic field build-ups as energy is transferred from the electric field to the
magnetic field, gradually the capacitor
loses
all of its charge and thus also loses
its electric field
and the energy stored in that field. The
energy
has now been fully transferred to the
magnetic field
of the inductor and the magnetic field
is then
maximum as you can see you here
and the current in the inductor in
this case is
maximum. Now although the capacitor is
completely discharged
the persists and doesn't become
zero
instantaneously and now the capacitor
begins
to get charged with a reverse polarity
opposite to that it had initially when we started.
now as the current decreases the magnetic field also decreases inducing an emf in
the inductor in the same
direction as the current
and this slows down the decrease of the current.
eventually, which is this stage, which we have
shown over here the current and the
magnetic field
read zero and now the capacitor is
charged with the opposite polarity
after this the same thing repeats
the capacitors starts getting discharged, its just that
in this case the polarity is reversed so the current will flow in this
direction
so the capacitor starts getting
discharged and the
and the magnetic field around the inductor starts building up
if i were to draw the current, it
becomes 0 in 3 which is this
situation over here after that
since a reverse polarity has built up
so this would be stage 4 when the current
is maximum in the opposite
direction
and you are back to original
stage which is stage five
so you can see that the current in the
circuit
is varying sinusoidally and we said that
the name is an
LC oscillator
We are going to write a differential equation
now
for the LC oscillator but it is
actually a little easier if you start
with the total energy in the system
so the total energy is the sum of the
energy in the electric
field plus the energy in the magnetic field, which can be written as
Differentiating total energy, U and putting it equal to 0 because U is a constant.
This is our differential equation and we are going to look for a solution
This is our differential equation and we are going to look for a solution
this differential equation has the same form as an equation which you have learnt earlier
earlier which is equation for a simple
harmonic oscillator
and that equation was
see if you don't recall the equation maybe you can just
go through it once
you have to start with F equal to
minus kx
because that is the restoring force then you
write F=ma
mass times
acceleration then you put the
acceleration as
a double differentiation of the
displacement
and then you substitute
w2=k/m
you get this equation which we have will
be here now
comparing these two equations we
can see that
this term and this term are similar
and hence the frequency of the circuit
is one over 2 pi root
LC and this is what we
wanted to find out
what is the frequency of this LC oscillator
frequency of the LC oscillator
is going to depend on the value of the
inductance
and the capacitance both
and just as for a simple harmonic
harmonic oscillator you have x = A cos
wt plus phi
similarly here we have q = q0
the amplitude the maximum charge in the
circuit
initially the capacitor was fully charged
the charge on the capacitor was maximum so
q was equal to q0 at time t =0
putting that into the equation here you can see that
the answers are discussed a short while
later